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<span>Mechanism</span> </div> </a> <ul id="toc-Mechanism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Examples</span> </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Examples subsection</span> </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Single-slit_diffraction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Single-slit_diffraction"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Single-slit diffraction</span> </div> </a> <ul id="toc-Single-slit_diffraction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diffraction_grating" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Diffraction_grating"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Diffraction grating</span> </div> </a> <ul id="toc-Diffraction_grating-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Circular_aperture" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Circular_aperture"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Circular aperture</span> </div> </a> <ul id="toc-Circular_aperture-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_aperture" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_aperture"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>General aperture</span> </div> </a> <ul id="toc-General_aperture-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Propagation_of_a_laser_beam" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Propagation_of_a_laser_beam"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Propagation of a laser beam</span> </div> </a> <ul id="toc-Propagation_of_a_laser_beam-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diffraction-limited_imaging" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Diffraction-limited_imaging"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Diffraction-limited imaging</span> </div> </a> <ul id="toc-Diffraction-limited_imaging-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Speckle_patterns" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Speckle_patterns"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.7</span> <span>Speckle patterns</span> </div> </a> <ul id="toc-Speckle_patterns-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Babinet's_principle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Babinet's_principle"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.8</span> <span>Babinet's principle</span> </div> </a> <ul id="toc-Babinet's_principle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-"Knife_edge"" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#"Knife_edge""> <div class="vector-toc-text"> <span class="vector-toc-numb">3.9</span> <span>"Knife edge"</span> </div> </a> <ul id="toc-"Knife_edge"-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Patterns" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Patterns"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Patterns</span> </div> </a> <ul id="toc-Patterns-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Matter_wave_diffraction" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Matter_wave_diffraction"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Matter wave diffraction</span> </div> </a> <ul id="toc-Matter_wave_diffraction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bragg_diffraction" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bragg_diffraction"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Bragg diffraction</span> </div> </a> <ul id="toc-Bragg_diffraction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Coherence" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Coherence"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Coherence</span> </div> </a> <ul id="toc-Coherence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Applications</span> </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Applications subsection</span> </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Diffraction_before_destruction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Diffraction_before_destruction"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Diffraction before destruction</span> </div> </a> <ul id="toc-Diffraction_before_destruction-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Diffraction</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 73 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-73" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">73 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8B%A8%E1%89%A5%E1%88%AD%E1%88%83%E1%8A%95_%E1%88%98%E1%8B%88%E1%88%8B%E1%8C%88%E1%8B%B5" title="የብርሃን መወላገድ – Amharic" lang="am" hreflang="am" data-title="የብርሃን መወላገድ" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AD%D9%8A%D9%88%D8%AF" title="حيود – Arabic" lang="ar" hreflang="ar" data-title="حيود" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Difraksiya" title="Difraksiya – Azerbaijani" lang="az" hreflang="az" data-title="Difraksiya" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A6%AA%E0%A6%AC%E0%A6%B0%E0%A7%8D%E0%A6%A4%E0%A6%A8" title="অপবর্তন – Bangla" lang="bn" hreflang="bn" data-title="অপবর্তন" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%94%D1%8B%D1%84%D1%80%D0%B0%D0%BA%D1%86%D1%8B%D1%8F" title="Дыфракцыя – Belarusian" lang="be" hreflang="be" data-title="Дыфракцыя" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D1%80%D0%B0%D0%BA%D1%86%D0%B8%D1%8F" title="Дифракция – Bulgarian" lang="bg" hreflang="bg" data-title="Дифракция" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Difracci%C3%B3" title="Difracció – Catalan" lang="ca" hreflang="ca" data-title="Difracció" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D1%80%D0%B0%D0%BA%D1%86%D0%B8" title="Дифракци – Chuvash" lang="cv" hreflang="cv" data-title="Дифракци" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Difrakce" title="Difrakce – Czech" lang="cs" hreflang="cs" data-title="Difrakce" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Bvurunuro" title="Bvurunuro – Shona" lang="sn" hreflang="sn" data-title="Bvurunuro" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Diffreithiant" title="Diffreithiant – Welsh" lang="cy" hreflang="cy" data-title="Diffreithiant" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Diffraktion" title="Diffraktion – Danish" lang="da" hreflang="da" data-title="Diffraktion" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Beugung_(Physik)" title="Beugung (Physik) – German" lang="de" hreflang="de" data-title="Beugung (Physik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Difraktsioon" title="Difraktsioon – Estonian" lang="et" hreflang="et" data-title="Difraktsioon" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%B5%CF%81%CE%AF%CE%B8%CE%BB%CE%B1%CF%83%CE%B7" title="Περίθλαση – Greek" lang="el" hreflang="el" data-title="Περίθλαση" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Difracci%C3%B3n" title="Difracción – Spanish" lang="es" hreflang="es" data-title="Difracción" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Difrakto" title="Difrakto – Esperanto" lang="eo" hreflang="eo" data-title="Difrakto" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Difrakzio_(fisika)" title="Difrakzio (fisika) – Basque" lang="eu" hreflang="eu" data-title="Difrakzio (fisika)" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%BE%D8%B1%D8%A7%D8%B4" title="پراش – Persian" lang="fa" hreflang="fa" data-title="پراش" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Diffraction" title="Diffraction – French" lang="fr" hreflang="fr" data-title="Diffraction" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/D%C3%ADraonadh" title="Díraonadh – Irish" lang="ga" hreflang="ga" data-title="Díraonadh" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Difracci%C3%B3n" title="Difracción – Galician" lang="gl" hreflang="gl" data-title="Difracción" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%9A%8C%EC%A0%88" title="회절 – Korean" lang="ko" hreflang="ko" data-title="회절" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B4%D5%AB%D6%86%D6%80%D5%A1%D5%AF%D6%81%D5%AB%D5%A1" title="Դիֆրակցիա – Armenian" lang="hy" hreflang="hy" data-title="Դիֆրակցիա" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%A4%E0%A4%A8" title="विवर्तन – Hindi" lang="hi" hreflang="hi" data-title="विवर्तन" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Ogib" title="Ogib – Croatian" lang="hr" hreflang="hr" data-title="Ogib" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Difraksi" title="Difraksi – Indonesian" lang="id" hreflang="id" data-title="Difraksi" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Diffrazione" title="Diffrazione – Italian" lang="it" hreflang="it" data-title="Diffrazione" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A2%D7%A7%D7%99%D7%A4%D7%94" title="עקיפה – Hebrew" lang="he" hreflang="he" data-title="עקיפה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D1%80%D0%B0%D0%BA%D1%86%D0%B8%D1%8F" title="Дифракция – Kazakh" lang="kk" hreflang="kk" data-title="Дифракция" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Difraksyon" title="Difraksyon – Haitian Creole" lang="ht" hreflang="ht" data-title="Difraksyon" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D1%80%D0%B0%D0%BA%D1%86%D0%B8%D1%8F" title="Дифракция – Kyrgyz" lang="ky" hreflang="ky" data-title="Дифракция" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Difrakcija" title="Difrakcija – Latvian" lang="lv" hreflang="lv" data-title="Difrakcija" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Difrakcija" title="Difrakcija – Lithuanian" lang="lt" hreflang="lt" data-title="Difrakcija" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Diffrazion" title="Diffrazion – Lombard" lang="lmo" hreflang="lmo" data-title="Diffrazion" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Diffrakci%C3%B3" title="Diffrakció – Hungarian" lang="hu" hreflang="hu" data-title="Diffrakció" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D1%80%D0%B0%D0%BA%D1%86%D0%B8%D1%98%D0%B0" title="Дифракција – Macedonian" lang="mk" hreflang="mk" data-title="Дифракција" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B5%E0%B4%BF%E0%B4%AD%E0%B4%82%E0%B4%97%E0%B4%A8%E0%B4%82" title="വിഭംഗനം – Malayalam" lang="ml" hreflang="ml" data-title="വിഭംഗനം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%A4%E0%A4%A8" title="विवर्तन – Marathi" lang="mr" hreflang="mr" data-title="विवर्तन" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Belauan" title="Belauan – Malay" lang="ms" hreflang="ms" data-title="Belauan" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D1%80%D0%B0%D0%BA%D1%86" title="Дифракц – Mongolian" lang="mn" hreflang="mn" data-title="Дифракц" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Diffractie" title="Diffractie – Dutch" lang="nl" hreflang="nl" data-title="Diffractie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%9B%9E%E6%8A%98" title="回折 – Japanese" lang="ja" hreflang="ja" data-title="回折" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Diffraksjon" title="Diffraksjon – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Diffraksjon" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Diffraksjon" title="Diffraksjon – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Diffraksjon" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Difraksiya" title="Difraksiya – Uzbek" lang="uz" hreflang="uz" data-title="Difraksiya" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B5%E0%A8%BF%E0%A8%B5%E0%A8%B0%E0%A8%A4%E0%A8%A8" title="ਵਿਵਰਤਨ – Punjabi" lang="pa" hreflang="pa" data-title="ਵਿਵਰਤਨ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Difrassion" title="Difrassion – Piedmontese" lang="pms" hreflang="pms" data-title="Difrassion" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Dyfrakcja" title="Dyfrakcja – Polish" lang="pl" hreflang="pl" data-title="Dyfrakcja" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Difra%C3%A7%C3%A3o" title="Difração – Portuguese" lang="pt" hreflang="pt" data-title="Difração" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Difrac%C8%9Bie" title="Difracție – Romanian" lang="ro" hreflang="ro" data-title="Difracție" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D1%80%D0%B0%D0%BA%D1%86%D0%B8%D1%8F" title="Дифракция – Russian" lang="ru" hreflang="ru" data-title="Дифракция" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Difraksioni_i_Drites" title="Difraksioni i Drites – Albanian" lang="sq" hreflang="sq" data-title="Difraksioni i Drites" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Diffrazzioni" title="Diffrazzioni – Sicilian" lang="scn" hreflang="scn" data-title="Diffrazzioni" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B7%80%E0%B7%92%E0%B7%80%E0%B6%BB%E0%B7%8A%E0%B6%AD%E0%B6%B1%E0%B6%BA" title="විවර්තනය – Sinhala" lang="si" hreflang="si" data-title="විවර්තනය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Diffraction" title="Diffraction – Simple English" lang="en-simple" hreflang="en-simple" data-title="Diffraction" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Difrakcia" title="Difrakcia – Slovak" lang="sk" hreflang="sk" data-title="Difrakcia" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Uklon" title="Uklon – Slovenian" lang="sl" hreflang="sl" data-title="Uklon" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D1%80%D0%B0%D0%BA%D1%86%D0%B8%D1%98%D0%B0" title="Дифракција – Serbian" lang="sr" hreflang="sr" data-title="Дифракција" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Difrakcija" title="Difrakcija – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Difrakcija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Difraksi" title="Difraksi – Sundanese" lang="su" hreflang="su" data-title="Difraksi" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Diffraktio" title="Diffraktio – Finnish" lang="fi" hreflang="fi" data-title="Diffraktio" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Diffraktion" title="Diffraktion – Swedish" lang="sv" hreflang="sv" data-title="Diffraktion" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AE%BF%E0%AE%B3%E0%AE%BF%E0%AE%AE%E0%AF%8D%E0%AE%AA%E0%AF%81_%E0%AE%B5%E0%AE%BF%E0%AE%B3%E0%AF%88%E0%AE%B5%E0%AF%81" title="விளிம்பு விளைவு – Tamil" lang="ta" hreflang="ta" data-title="விளிம்பு விளைவு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%B5%E0%B0%BF%E0%B0%B5%E0%B0%B0%E0%B1%8D%E0%B0%A4%E0%B0%A8%E0%B0%82" title="వివర్తనం – Telugu" lang="te" hreflang="te" data-title="వివర్తనం" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%80%E0%B8%A5%E0%B8%B5%E0%B9%89%E0%B8%A2%E0%B8%A7%E0%B9%80%E0%B8%9A%E0%B8%99" title="การเลี้ยวเบน – Thai" lang="th" hreflang="th" data-title="การเลี้ยวเบน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/K%C4%B1r%C4%B1n%C4%B1m" title="Kırınım – Turkish" lang="tr" hreflang="tr" data-title="Kırınım" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D1%80%D0%B0%D0%BA%D1%86%D1%96%D1%8F" title="Дифракція – Ukrainian" lang="uk" hreflang="uk" data-title="Дифракція" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%A7%D9%86%DA%A9%D8%B3%D8%A7%D8%B1_%D9%86%D9%88%D8%B1" title="انکسار نور – Urdu" lang="ur" hreflang="ur" data-title="انکسار نور" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Nhi%E1%BB%85u_x%E1%BA%A1" title="Nhiễu xạ – Vietnamese" lang="vi" hreflang="vi" data-title="Nhiễu xạ" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E8%A1%8D%E5%B0%84" title="衍射 – Wu" lang="wuu" hreflang="wuu" data-title="衍射" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%B9%9E%E5%B0%84" title="繞射 – Cantonese" lang="yue" hreflang="yue" data-title="繞射" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://zh.wikipedia.org/wiki/%E8%A1%8D%E5%B0%84" title="衍射 – Chinese" lang="zh" hreflang="zh" data-title="衍射" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q133900#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> 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data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Refraction" title="Refraction">refraction</a>, the change in direction of a wave passing from one medium to another.</div> <p class="mw-empty-elt"> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Laser_Interference.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Laser_Interference.JPG/220px-Laser_Interference.JPG" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Laser_Interference.JPG/330px-Laser_Interference.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/42/Laser_Interference.JPG/440px-Laser_Interference.JPG 2x" data-file-width="1950" data-file-height="1950" /></a><figcaption>A <a href="/wiki/Airy_disk" title="Airy disk">diffraction pattern</a> of a red <a href="/wiki/Laser" title="Laser">laser</a> beam projected onto a plate after passing through a small circular <a href="/wiki/Aperture" title="Aperture">aperture</a> in another plate</figcaption></figure> <p><b>Diffraction</b> is the interference or bending of waves around the corners of an obstacle or through an <a href="/wiki/Aperture" title="Aperture">aperture</a> into the region of geometrical <a href="/wiki/Shadow" title="Shadow">shadow</a> of the obstacle/aperture. The diffracting object or aperture effectively becomes a secondary source of the <a href="/wiki/Wave_propagation" class="mw-redirect" title="Wave propagation">propagating</a> wave. Italian scientist <a href="/wiki/Francesco_Maria_Grimaldi" title="Francesco Maria Grimaldi">Francesco Maria Grimaldi</a> coined the word <i>diffraction</i> and was the first to record accurate observations of the phenomenon in <a href="/wiki/1660_in_science" title="1660 in science">1660</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Single_Slit_Diffraction.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Single_Slit_Diffraction.svg/260px-Single_Slit_Diffraction.svg.png" decoding="async" width="260" height="223" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Single_Slit_Diffraction.svg/390px-Single_Slit_Diffraction.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Single_Slit_Diffraction.svg/520px-Single_Slit_Diffraction.svg.png 2x" data-file-width="700" data-file-height="600" /></a><figcaption>Infinitely many points (three shown) along length <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> project phase contributions from the <a href="/wiki/Wavefront" title="Wavefront">wavefront</a>, producing a continuously varying intensity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> on the registering plate</figcaption></figure> <p>In <a href="/wiki/Classical_physics" title="Classical physics">classical physics</a>, the diffraction phenomenon is described by the <a href="/wiki/Huygens%E2%80%93Fresnel_principle" title="Huygens–Fresnel principle">Huygens–Fresnel principle</a> that treats each point in a propagating <a href="/wiki/Wavefront" title="Wavefront">wavefront</a> as a collection of individual spherical <a href="/wiki/Wavelet" title="Wavelet">wavelets</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The characteristic bending pattern is most pronounced when a wave from a <a href="/wiki/Coherence_(physics)" title="Coherence (physics)">coherent</a> source (such as a laser) encounters a slit/aperture that is comparable in size to its <a href="/wiki/Wavelength" title="Wavelength">wavelength</a>, as shown in the inserted image. This is due to the addition, or <a href="/wiki/Interference_(wave_propagation)" class="mw-redirect" title="Interference (wave propagation)">interference</a>, of different points on the wavefront (or, equivalently, each wavelet) that travel by paths of different lengths to the registering surface. If there are multiple, <a href="/wiki/Diffraction_grating" title="Diffraction grating">closely spaced openings</a> (e.g., a <a href="/wiki/Diffraction_grating" title="Diffraction grating">diffraction grating</a>), a complex pattern of varying intensity can result. </p><p>These effects also occur when a <a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">light wave</a> travels through a medium with a varying <a href="/wiki/Refractive_index" title="Refractive index">refractive index</a>, or when a <a href="/wiki/Sound" title="Sound">sound wave</a> travels through a medium with varying <a href="/wiki/Acoustic_impedance" title="Acoustic impedance">acoustic impedance</a> – all waves diffract,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> including <a href="/wiki/Gravitational_wave" title="Gravitational wave">gravitational waves</a>,<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Wind_wave" title="Wind wave">water waves</a>, and other <a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">electromagnetic waves</a> such as <a href="/wiki/X-ray" title="X-ray">X-rays</a> and <a href="/wiki/Radio_waves" class="mw-redirect" title="Radio waves">radio waves</a>. Furthermore, <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> also demonstrates that matter possesses <a href="/wiki/Matter_wave" title="Matter wave">wave-like properties</a> and, therefore, undergoes diffraction (which is measurable at subatomic to molecular levels).<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>The amount of diffraction depends on the size of the gap. Diffraction is greatest when the size of the gap is similar to the wavelength of the wave. In this case, when the waves pass through the gap they become <a href="/wiki/Semicircle" title="Semicircle">semi-circular</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffraction&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Young_Diffraction.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Young_Diffraction.png/220px-Young_Diffraction.png" decoding="async" width="220" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Young_Diffraction.png/330px-Young_Diffraction.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Young_Diffraction.png/440px-Young_Diffraction.png 2x" data-file-width="1988" data-file-height="882" /></a><figcaption>Thomas Young's sketch of two-slit diffraction for water waves, which he presented to the Royal Society in <a href="/wiki/1803_in_science" title="1803 in science">1803</a></figcaption></figure> <p><a href="/wiki/Leonardo_da_Vinci" title="Leonardo da Vinci">Da Vinci</a> might have observed diffraction in a broadening of the shadow.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> The effects of diffraction of light were first carefully observed and characterized by <a href="/wiki/Francesco_Maria_Grimaldi" title="Francesco Maria Grimaldi">Francesco Maria Grimaldi</a>, who also coined the term <i>diffraction</i>, from the <a href="/wiki/Latin" title="Latin">Latin</a> <i>diffringere</i>, 'to break into pieces', referring to light breaking up into different directions. The results of Grimaldi's observations were published posthumously in <a href="/wiki/1665_in_science" title="1665 in science">1665</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> studied these effects and attributed them to <i>inflexion</i> of light rays. <a href="/wiki/James_Gregory_(astronomer_and_mathematician)" class="mw-redirect" title="James Gregory (astronomer and mathematician)">James Gregory</a> (<a href="/wiki/1638" title="1638">1638</a>–<a href="/wiki/1675" title="1675">1675</a>) observed the diffraction patterns caused by a bird feather, which was effectively the first <a href="/wiki/Diffraction_grating" title="Diffraction grating">diffraction grating</a> to be discovered.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Thomas_Young_(scientist)" title="Thomas Young (scientist)">Thomas Young</a> performed a <a href="/wiki/Young%27s_interference_experiment" title="Young's interference experiment">celebrated experiment</a> in <a href="/wiki/1803_in_science" title="1803 in science">1803</a> demonstrating interference from two closely spaced slits.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Explaining his results by interference of the waves emanating from the two different slits, he deduced that light must propagate as waves. <a href="/wiki/Augustin-Jean_Fresnel" title="Augustin-Jean Fresnel">Augustin-Jean Fresnel</a> did more definitive studies and calculations of diffraction, made public in <a href="/wiki/1816_in_science" title="1816 in science">1816</a><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/1818_in_science" title="1818 in science">1818</a>,<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> and thereby gave great support to the wave theory of light that had been advanced by <a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Christiaan Huygens</a><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> and reinvigorated by Young, against Newton's <a href="/wiki/Corpuscular_theory_of_light" title="Corpuscular theory of light">corpuscular theory of light</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Mechanism">Mechanism</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffraction&action=edit&section=2" title="Edit section: Mechanism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Single-slit-diffraction-ripple-tank.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Single-slit-diffraction-ripple-tank.jpg/220px-Single-slit-diffraction-ripple-tank.jpg" decoding="async" width="220" height="146" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Single-slit-diffraction-ripple-tank.jpg/330px-Single-slit-diffraction-ripple-tank.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Single-slit-diffraction-ripple-tank.jpg/440px-Single-slit-diffraction-ripple-tank.jpg 2x" data-file-width="1544" data-file-height="1024" /></a><figcaption>Single-slit diffraction in a circular <a href="/wiki/Ripple_tank" title="Ripple tank">ripple tank</a></figcaption></figure> <p>In <a href="/wiki/Classical_physics" title="Classical physics">classical physics</a> diffraction arises because of how <a href="/wiki/Wave" title="Wave">waves</a> propagate; this is described by the <a href="/wiki/Huygens%E2%80%93Fresnel_principle" title="Huygens–Fresnel principle">Huygens–Fresnel principle</a> and the <a href="/wiki/Superposition_principle" title="Superposition principle">principle of superposition of waves</a>. The propagation of a wave can be visualized by considering every particle of the transmitted medium on a wavefront as a <a href="/wiki/Point_source" title="Point source">point source</a> for a secondary <a href="/wiki/Wave_equation#Spherical_waves" title="Wave equation">spherical wave</a>. The wave displacement at any subsequent point is the sum of these secondary waves. When waves are added together, their sum is determined by the relative <a href="/wiki/Phase_(waves)" title="Phase (waves)">phases</a> as well as the <a href="/wiki/Amplitude" title="Amplitude">amplitudes</a> of the individual waves so that the summed amplitude of the waves can have any value between zero and the sum of the individual amplitudes. Hence, diffraction patterns usually have a series of maxima and minima. </p><p>In the <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">modern quantum mechanical</a> understanding of light propagation through a slit (or slits) every <a href="/wiki/Photon" title="Photon">photon</a> is described by its <a href="/wiki/Wavefunction" class="mw-redirect" title="Wavefunction">wavefunction</a> that determines the <a href="/wiki/Probability_amplitude" title="Probability amplitude">probability distribution</a> for the photon: the light and dark bands are the areas where the photons are more or less likely to be detected. The wavefunction is determined by the physical surroundings such as slit geometry, screen distance, and initial conditions when the photon is created. The wave nature of individual photons (as opposed to wave properties only arising from the interactions between multitudes of photons) was implied by a low-intensity <a href="/wiki/Double-slit_experiment" title="Double-slit experiment">double-slit experiment</a> first performed by <a href="/wiki/G._I._Taylor" title="G. I. Taylor">G. I. Taylor</a> in <a href="/wiki/1909_in_science" title="1909 in science">1909</a>. The quantum approach has some striking similarities to the <a href="/wiki/Huygens-Fresnel_principle" class="mw-redirect" title="Huygens-Fresnel principle">Huygens-Fresnel principle</a>; based on that principle, as light travels through slits and boundaries, secondary point light sources are created near or along these obstacles, and the resulting diffraction pattern is going to be the intensity profile based on the collective interference of all these light sources that have different optical paths. In the quantum formalism, that is similar to considering the limited regions around the slits and boundaries from which photons are more likely to originate, and calculating the probability distribution (that is proportional to the resulting intensity of classical formalism). </p><p>There are various analytical models which allow the diffracted field to be calculated, including the <a href="/wiki/Kirchhoff%27s_diffraction_formula" title="Kirchhoff's diffraction formula">Kirchhoff diffraction equation</a> (derived from the <a href="/wiki/Wave_equation" title="Wave equation">wave equation</a>),<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> the <a href="/wiki/Fraunhofer_diffraction" title="Fraunhofer diffraction">Fraunhofer diffraction</a> approximation of the Kirchhoff equation (applicable to the <a href="/wiki/Near_and_far_field#Far_field" title="Near and far field">far field</a>), the <a href="/wiki/Fresnel_diffraction" title="Fresnel diffraction">Fresnel diffraction</a> approximation (applicable to the <a href="/wiki/Near_and_far_field#Near_field" title="Near and far field">near field</a>) and the Feynman <a href="/wiki/Path_integral_formulation" title="Path integral formulation">path integral formulation</a>. Most configurations cannot be solved analytically, but can yield numerical solutions through <a href="/wiki/Finite_element" class="mw-redirect" title="Finite element">finite element</a> and <a href="/wiki/Boundary_element" class="mw-redirect" title="Boundary element">boundary element</a> methods. </p><p>It is possible to obtain a qualitative understanding of many diffraction phenomena by considering how the relative phases of the individual secondary wave sources vary, and, in particular, the conditions in which the phase difference equals half a cycle in which case waves will cancel one another out. </p><p>The simplest descriptions of diffraction are those in which the situation can be reduced to a two-dimensional problem. For <a href="/wiki/Wind_wave" title="Wind wave">water waves</a>, this is already the case; water waves propagate only on the surface of the water. For light, we can often neglect one direction if the diffracting object extends in that direction over a distance far greater than the wavelength. In the case of light shining through small circular holes, we will have to take into account the full three-dimensional nature of the problem. </p> <ul class="gallery mw-gallery-packed"> <li class="gallerybox" style="width: 122px"> <div class="thumb" style="width: 120px;"><span typeof="mw:File"><a href="/wiki/File:Square_diffraction.jpg" class="mw-file-description" title="Computer-generated intensity pattern formed on a screen by diffraction from a square aperture"><img alt="Computer-generated intensity pattern formed on a screen by diffraction from a square aperture" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Square_diffraction.jpg/180px-Square_diffraction.jpg" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Square_diffraction.jpg/269px-Square_diffraction.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Square_diffraction.jpg/359px-Square_diffraction.jpg 2x" data-file-width="430" data-file-height="431" /></a></span></div> <div class="gallerytext">Computer-generated intensity pattern formed on a screen by diffraction from a square aperture</div> </li> <li class="gallerybox" style="width: 152px"> <div class="thumb" style="width: 150px;"><span typeof="mw:File"><a href="/wiki/File:Two-Slit_Diffraction.png" class="mw-file-description" title="Generation of an interference pattern from two-slit diffraction"><img alt="Generation of an interference pattern from two-slit diffraction" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/Two-Slit_Diffraction.png/225px-Two-Slit_Diffraction.png" decoding="async" width="150" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/Two-Slit_Diffraction.png/338px-Two-Slit_Diffraction.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/01/Two-Slit_Diffraction.png/450px-Two-Slit_Diffraction.png 2x" data-file-width="1280" data-file-height="1024" /></a></span></div> <div class="gallerytext">Generation of an interference pattern from two-slit diffraction</div> </li> <li class="gallerybox" style="width: 123.33333333333px"> <div class="thumb" style="width: 121.33333333333px;"><span typeof="mw:File"><a href="/wiki/File:Doubleslit.gif" class="mw-file-description" title="Computational model of an interference pattern from two-slit diffraction"><img alt="Computational model of an interference pattern from two-slit diffraction" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Doubleslit.gif/182px-Doubleslit.gif" decoding="async" width="122" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Doubleslit.gif/273px-Doubleslit.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/a/a9/Doubleslit.gif 2x" data-file-width="304" data-file-height="301" /></a></span></div> <div class="gallerytext">Computational model of an interference pattern from two-slit diffraction</div> </li> <li class="gallerybox" style="width: 152.66666666667px"> <div class="thumb" style="width: 150.66666666667px;"><span typeof="mw:File"><a href="/wiki/File:Optical_diffraction_pattern_(_laser),_(analogous_to_X-ray_crystallography).JPG" class="mw-file-description" title="Optical diffraction pattern (laser, analogous to X-ray diffraction)"><img alt="Optical diffraction pattern (laser, analogous to X-ray diffraction)" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Optical_diffraction_pattern_%28_laser%29%2C_%28analogous_to_X-ray_crystallography%29.JPG/226px-Optical_diffraction_pattern_%28_laser%29%2C_%28analogous_to_X-ray_crystallography%29.JPG" decoding="async" width="151" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Optical_diffraction_pattern_%28_laser%29%2C_%28analogous_to_X-ray_crystallography%29.JPG/340px-Optical_diffraction_pattern_%28_laser%29%2C_%28analogous_to_X-ray_crystallography%29.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Optical_diffraction_pattern_%28_laser%29%2C_%28analogous_to_X-ray_crystallography%29.JPG/453px-Optical_diffraction_pattern_%28_laser%29%2C_%28analogous_to_X-ray_crystallography%29.JPG 2x" data-file-width="855" data-file-height="680" /></a></span></div> <div class="gallerytext">Optical diffraction pattern (laser, analogous to X-ray diffraction)</div> </li> <li class="gallerybox" style="width: 182px"> <div class="thumb" style="width: 180px;"><span typeof="mw:File"><a href="/wiki/File:Diffraction_pattern_in_spiderweb.JPG" class="mw-file-description" title="Colors seen in a spider web are partially due to diffraction, according to some analyses.[17]"><img alt="Colors seen in a spider web are partially due to diffraction, according to some analyses.[17]" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Diffraction_pattern_in_spiderweb.JPG/270px-Diffraction_pattern_in_spiderweb.JPG" decoding="async" width="180" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Diffraction_pattern_in_spiderweb.JPG/405px-Diffraction_pattern_in_spiderweb.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Diffraction_pattern_in_spiderweb.JPG/540px-Diffraction_pattern_in_spiderweb.JPG 2x" data-file-width="3888" data-file-height="2592" /></a></span></div> <div class="gallerytext">Colors seen in a <a href="/wiki/Spider_web" title="Spider web">spider web</a> are partially due to diffraction, according to some analyses.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup></div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffraction&action=edit&section=3" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The effects of diffraction are often seen in everyday life. The most striking examples of diffraction are those that involve light; for example, the closely spaced tracks on a CD or DVD act as a <a href="/wiki/Diffraction_grating" title="Diffraction grating">diffraction grating</a> to form the familiar rainbow pattern seen when looking at a disc. </p> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tnone center"><div class="thumbinner multiimageinner" style="width:408px;max-width:408px"><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Screendiffraction.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Screendiffraction.jpg/200px-Screendiffraction.jpg" decoding="async" width="200" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Screendiffraction.jpg/300px-Screendiffraction.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Screendiffraction.jpg/400px-Screendiffraction.jpg 2x" data-file-width="1280" data-file-height="960" /></a></span></div><div class="thumbcaption">Pixels on smart phone screen acting as diffraction grating</div></div><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Sunlight_diffraction_off_of_cd_rom.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Sunlight_diffraction_off_of_cd_rom.jpg/200px-Sunlight_diffraction_off_of_cd_rom.jpg" decoding="async" width="200" height="141" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Sunlight_diffraction_off_of_cd_rom.jpg/300px-Sunlight_diffraction_off_of_cd_rom.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Sunlight_diffraction_off_of_cd_rom.jpg/400px-Sunlight_diffraction_off_of_cd_rom.jpg 2x" data-file-width="929" data-file-height="657" /></a></span></div><div class="thumbcaption">Data is written on CDs as pits and lands; the pits on the surface act as diffracting elements.</div></div></div></div></div> <p>This principle can be extended to engineer a grating with a structure such that it will produce any diffraction pattern desired; the <a href="/wiki/Holography" title="Holography">hologram</a> on a credit card is an example. </p><p><a href="/wiki/Atmospheric_diffraction" title="Atmospheric diffraction">Diffraction in the atmosphere</a> by small particles can cause a <a href="/wiki/Corona_(optical_phenomenon)" title="Corona (optical phenomenon)">corona</a> - a bright disc and rings around a bright light source like the sun or the moon. At the opposite point one may also observe <a href="/wiki/Glory_(optical_phenomenon)" title="Glory (optical phenomenon)">glory</a> - bright rings around the shadow of the observer. In contrast to the corona, glory requires the particles to be transparent spheres (like fog droplets), since the <a href="/wiki/Backscatter" title="Backscatter">backscattering</a> of the light that forms the glory involves <a href="/wiki/Refraction" title="Refraction">refraction</a> and internal reflection within the droplet. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tnone center"><div class="thumbinner multiimageinner" style="width:408px;max-width:408px"><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Lunar_Halo_.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Lunar_Halo_.jpg/200px-Lunar_Halo_.jpg" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Lunar_Halo_.jpg/300px-Lunar_Halo_.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/30/Lunar_Halo_.jpg/400px-Lunar_Halo_.jpg 2x" data-file-width="2448" data-file-height="2448" /></a></span></div><div class="thumbcaption">Lunar <a href="/wiki/Corona_(optical_phenomenon)" title="Corona (optical phenomenon)">corona</a>.</div></div><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:IMG_7474_solar_glory.JPG" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/IMG_7474_solar_glory.JPG/200px-IMG_7474_solar_glory.JPG" decoding="async" width="200" height="133" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/IMG_7474_solar_glory.JPG/300px-IMG_7474_solar_glory.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/IMG_7474_solar_glory.JPG/400px-IMG_7474_solar_glory.JPG 2x" data-file-width="3456" data-file-height="2304" /></a></span></div><div class="thumbcaption">A solar <a href="/wiki/Glory_(optical_phenomenon)" title="Glory (optical phenomenon)">glory</a>, as seen from a plane on the underlying clouds.</div></div></div></div></div> <p>A shadow of a solid object, using light from a compact source, shows small fringes near its edges. </p> <figure class="mw-default-size mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Poissonspot_simulation_d4mm.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Poissonspot_simulation_d4mm.jpg/220px-Poissonspot_simulation_d4mm.jpg" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Poissonspot_simulation_d4mm.jpg/330px-Poissonspot_simulation_d4mm.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Poissonspot_simulation_d4mm.jpg/440px-Poissonspot_simulation_d4mm.jpg 2x" data-file-width="800" data-file-height="800" /></a><figcaption>The bright spot (<a href="/wiki/Arago_spot" title="Arago spot">Arago spot</a>) seen in the center of the shadow of a circular obstacle is due to diffraction</figcaption></figure> <p><a href="/wiki/Diffraction_spikes" class="mw-redirect" title="Diffraction spikes">Diffraction spikes</a> are diffraction patterns caused due to non-circular <a href="/wiki/Aperture" title="Aperture">aperture</a> in camera or support struts in telescope; In normal vision, diffraction through eyelashes may produce such spikes. </p> <figure class="mw-default-size mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Night_London_Panorama_with_Full_Moon.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Night_London_Panorama_with_Full_Moon.jpg/220px-Night_London_Panorama_with_Full_Moon.jpg" decoding="async" width="220" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Night_London_Panorama_with_Full_Moon.jpg/330px-Night_London_Panorama_with_Full_Moon.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/Night_London_Panorama_with_Full_Moon.jpg/440px-Night_London_Panorama_with_Full_Moon.jpg 2x" data-file-width="800" data-file-height="450" /></a><figcaption>View from the end of Millennium Bridge; Moon rising above the Southwark Bridge. Street lights are reflecting in the Thames.</figcaption></figure> <figure class="mw-default-size mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:FOFC8ZPX0AIB-Ho.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/FOFC8ZPX0AIB-Ho.png/220px-FOFC8ZPX0AIB-Ho.png" decoding="async" width="220" height="99" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/FOFC8ZPX0AIB-Ho.png/330px-FOFC8ZPX0AIB-Ho.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/FOFC8ZPX0AIB-Ho.png/440px-FOFC8ZPX0AIB-Ho.png 2x" data-file-width="1280" data-file-height="576" /></a><figcaption>Simulated diffraction spikes in hexagonal telescope mirrors</figcaption></figure> <p>The <a href="/wiki/Speckle_pattern" class="mw-redirect" title="Speckle pattern">speckle pattern</a> which is observed when laser light falls on an optically rough surface is also a diffraction phenomenon. When <a href="/wiki/Deli_meat" class="mw-redirect" title="Deli meat">deli meat</a> appears to be <a href="/wiki/Iridescent" class="mw-redirect" title="Iridescent">iridescent</a>, that is diffraction off the meat fibers.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> All these effects are a consequence of the fact that light propagates as a <a href="/wiki/Wave" title="Wave">wave</a>. </p><p> Diffraction can occur with any kind of wave. Ocean waves diffract around <a href="/wiki/Jetty" title="Jetty">jetties</a> and other obstacles. </p><figure class="mw-default-size mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Wave_diffraction_at_the_Blue_Lagoon,_Abereiddy.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Wave_diffraction_at_the_Blue_Lagoon%2C_Abereiddy.jpg/220px-Wave_diffraction_at_the_Blue_Lagoon%2C_Abereiddy.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Wave_diffraction_at_the_Blue_Lagoon%2C_Abereiddy.jpg/330px-Wave_diffraction_at_the_Blue_Lagoon%2C_Abereiddy.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Wave_diffraction_at_the_Blue_Lagoon%2C_Abereiddy.jpg/440px-Wave_diffraction_at_the_Blue_Lagoon%2C_Abereiddy.jpg 2x" data-file-width="4032" data-file-height="3024" /></a><figcaption>Circular waves generated by diffraction from the narrow entrance of a flooded coastal quarry</figcaption></figure><p> Sound waves can diffract around objects, which is why one can still hear someone calling even when hiding behind a tree.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>Diffraction can also be a concern in some technical applications; it sets a <a href="/wiki/Diffraction-limited_system" title="Diffraction-limited system">fundamental limit</a> to the resolution of a camera, telescope, or microscope. </p><p>Other examples of diffraction are considered below. </p> <div class="mw-heading mw-heading3"><h3 id="Single-slit_diffraction">Single-slit diffraction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffraction&action=edit&section=4" title="Edit section: Single-slit diffraction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Diffraction_formalism" class="mw-redirect" title="Diffraction formalism">Diffraction formalism</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:DiffractionSingleSlit_Anim.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/DiffractionSingleSlit_Anim.gif/220px-DiffractionSingleSlit_Anim.gif" decoding="async" width="220" height="203" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/DiffractionSingleSlit_Anim.gif/330px-DiffractionSingleSlit_Anim.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/82/DiffractionSingleSlit_Anim.gif/440px-DiffractionSingleSlit_Anim.gif 2x" data-file-width="619" data-file-height="572" /></a><figcaption>2D Single-slit diffraction with width changing animation</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Wave_Diffraction_4Lambda_Slit.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Wave_Diffraction_4Lambda_Slit.png/220px-Wave_Diffraction_4Lambda_Slit.png" decoding="async" width="220" height="177" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Wave_Diffraction_4Lambda_Slit.png/330px-Wave_Diffraction_4Lambda_Slit.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Wave_Diffraction_4Lambda_Slit.png/440px-Wave_Diffraction_4Lambda_Slit.png 2x" data-file-width="620" data-file-height="500" /></a><figcaption>Numerical approximation of diffraction pattern from a slit of width four wavelengths with an incident plane wave. The main central beam, nulls, and phase reversals are apparent.</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Single_Slit_Diffraction_(english).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Single_Slit_Diffraction_%28english%29.svg/220px-Single_Slit_Diffraction_%28english%29.svg.png" decoding="async" width="220" height="109" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Single_Slit_Diffraction_%28english%29.svg/330px-Single_Slit_Diffraction_%28english%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Single_Slit_Diffraction_%28english%29.svg/440px-Single_Slit_Diffraction_%28english%29.svg.png 2x" data-file-width="434" data-file-height="215" /></a><figcaption>Graph and image of single-slit diffraction</figcaption></figure> <p>A long slit of infinitesimal width which is illuminated by light diffracts the light into a series of circular waves and the wavefront which emerges from the slit is a cylindrical wave of uniform intensity, in accordance with the <a href="/wiki/Huygens%E2%80%93Fresnel_principle" title="Huygens–Fresnel principle">Huygens–Fresnel principle</a>. </p><p>An illuminated slit that is wider than a wavelength produces interference effects in the space downstream of the slit. Assuming that the slit behaves as though it has a large number of point sources spaced evenly across the width of the slit interference effects can be calculated. The analysis of this system is simplified if we consider light of a single wavelength. If the incident light is <a href="/wiki/Coherence_(physics)#Examples" title="Coherence (physics)">coherent</a>, these sources all have the same phase. Light incident at a given point in the space downstream of the slit is made up of contributions from each of these point sources and if the relative phases of these contributions vary by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }"></span> or more, we may expect to find minima and maxima in the diffracted light. Such phase differences are caused by differences in the path lengths over which contributing rays reach the point from the slit. </p><p>We can find the angle at which a first minimum is obtained in the diffracted light by the following reasoning. The light from a source located at the top edge of the slit interferes destructively with a source located at the middle of the slit, when the path difference between them is equal to <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda /2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f189d538c95274dd5297140e5ee78df90faf13e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.68ex; height:2.843ex;" alt="{\displaystyle \lambda /2}"></span>.</span> Similarly, the source just below the top of the slit will interfere destructively with the source located just below the middle of the slit at the same angle. We can continue this reasoning along the entire height of the slit to conclude that the condition for destructive interference for the entire slit is the same as the condition for destructive interference between two narrow slits a distance apart that is half the width of the slit. The path difference is approximately <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d\sin(\theta )}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\sin(\theta )}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1d8546190f9e2e03018928fcf6fa55586a79921" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.194ex; height:5.676ex;" alt="{\displaystyle {\frac {d\sin(\theta )}{2}}}"></span> so that the minimum intensity occurs at an angle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{\text{min}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>min</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{\text{min}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc9e738a4d02bf0842f046fe40254b19a07ffa26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.063ex; height:2.509ex;" alt="{\displaystyle \theta _{\text{min}}}"></span> given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\,\sin \theta _{\text{min}}=\lambda ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mspace width="thinmathspace" /> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>min</mtext> </mrow> </msub> <mo>=</mo> <mi>λ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\,\sin \theta _{\text{min}}=\lambda ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15f6a12801d8a1eefdfd9db64bb7728101bba0e3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.396ex; height:2.509ex;" alt="{\displaystyle d\,\sin \theta _{\text{min}}=\lambda ,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> is the width of the slit, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{\text{min}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>min</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{\text{min}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc9e738a4d02bf0842f046fe40254b19a07ffa26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.063ex; height:2.509ex;" alt="{\displaystyle \theta _{\text{min}}}"></span> is the <a href="/wiki/Angle_of_incidence_(optics)" title="Angle of incidence (optics)">angle of incidence</a> at which the minimum intensity occurs, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> is the wavelength of the light. </p><p>A similar argument can be used to show that if we imagine the slit to be divided into four, six, eight parts, etc., minima are obtained at angles <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79cc00920259451fc1a684ba7350b6f93ce4f08a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.309ex; height:2.509ex;" alt="{\displaystyle \theta _{n}}"></span> given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\,\sin \theta _{n}=n\lambda ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mspace width="thinmathspace" /> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>n</mi> <mi>λ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\,\sin \theta _{n}=n\lambda ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81d3aa9eba7f7c02b8af852e6f63fe5ed182e38b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.037ex; height:2.509ex;" alt="{\displaystyle d\,\sin \theta _{n}=n\lambda ,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is an integer other than zero. </p><p>There is no such simple argument to enable us to find the maxima of the diffraction pattern. The <a href="/wiki/Diffraction_formalism#Quantitative_analysis_of_single-slit_diffraction" class="mw-redirect" title="Diffraction formalism">intensity profile</a> can be calculated using the <a href="/wiki/Fraunhofer_diffraction" title="Fraunhofer diffraction">Fraunhofer diffraction</a> equation as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(\theta )=I_{0}\,\operatorname {sinc} ^{2}\left({\frac {d\pi }{\lambda }}\sin \theta \right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>sinc</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>π</mi> </mrow> <mi>λ</mi> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(\theta )=I_{0}\,\operatorname {sinc} ^{2}\left({\frac {d\pi }{\lambda }}\sin \theta \right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0362aa8af24f7af1665b3209427c1305b5edc512" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.136ex; height:6.176ex;" alt="{\displaystyle I(\theta )=I_{0}\,\operatorname {sinc} ^{2}\left({\frac {d\pi }{\lambda }}\sin \theta \right),}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e3621a13a1f446a54ba41c15f7ad7e164099c0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.071ex; height:2.843ex;" alt="{\displaystyle I(\theta )}"></span> is the intensity at a given angle, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/893d08e90ea73781dc133414d661529d0651ca80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.077ex; height:2.509ex;" alt="{\displaystyle I_{0}}"></span> is the intensity at the central maximum <span class="nowrap">(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a7bc6e34b53e0e8a8815159c356b1acccf7ea24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle \theta =0}"></span>),</span> which is also a normalization factor of the intensity profile that can be determined by an integration from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \theta =-{\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \theta =-{\frac {\pi }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97be652d7e553c7512d00db6c71c8a17fb42c383" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.775ex; height:3.176ex;" alt="{\textstyle \theta =-{\frac {\pi }{2}}}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \theta ={\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \theta ={\frac {\pi }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e6095ebb8d0b6505f453ae2da2fbd3477ecd3a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.967ex; height:3.176ex;" alt="{\textstyle \theta ={\frac {\pi }{2}}}"></span> and conservation of energy, and <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sinc} x={\frac {\sin x}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sinc</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {sinc} x={\frac {\sin x}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38219706f12b6e0957f539edd2a9a6b2867de92f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.112ex; height:5.176ex;" alt="{\displaystyle \operatorname {sinc} x={\frac {\sin x}{x}}}"></span>,</span> which is the <a href="/wiki/Unnormalized_sinc_function" class="mw-redirect" title="Unnormalized sinc function">unnormalized sinc function</a>. </p><p>This analysis applies only to the <a href="/wiki/Far_field" class="mw-redirect" title="Far field">far field</a> (<a href="/wiki/Fraunhofer_diffraction" title="Fraunhofer diffraction">Fraunhofer diffraction</a>), that is, at a distance much larger than the width of the slit. </p><p>From the <a href="/wiki/Diffraction_formalism#Quantitative_analysis_of_single-slit_diffraction" class="mw-redirect" title="Diffraction formalism">intensity profile</a> above, if <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\ll \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>≪</mo> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\ll \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b36d3e6b23030fc022f26c2e8b5775684b381858" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.185ex; height:2.176ex;" alt="{\displaystyle d\ll \lambda }"></span>,</span> the intensity will have little dependency on <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span>,</span> hence the wavefront emerging from the slit would resemble a cylindrical wave with azimuthal symmetry; If <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\gg \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>≫</mo> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\gg \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44957827facb1948efd78308454b7783346d0a76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.185ex; height:2.176ex;" alt="{\displaystyle d\gg \lambda }"></span>,</span> only <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta \approx 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>≈</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta \approx 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45fa623dea25165982c15695fcc75fe61d9bd404" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle \theta \approx 0}"></span> would have appreciable intensity, hence the wavefront emerging from the slit would resemble that of <a href="/wiki/Geometrical_optics" title="Geometrical optics">geometrical optics</a>. </p><p>When the incident angle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{\text{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{\text{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7f1cbab9315aa0f5290e429d9b2366f560dc39f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.78ex; height:2.509ex;" alt="{\displaystyle \theta _{\text{i}}}"></span> of the light onto the slit is non-zero (which causes a change in the <a href="/wiki/Optical_path_length" title="Optical path length">path length</a>), the intensity profile in the Fraunhofer regime (i.e. far field) becomes: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(\theta )=I_{0}\,\operatorname {sinc} ^{2}\left[{\frac {d\pi }{\lambda }}(\sin \theta \pm \sin \theta _{\text{i}})\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>sinc</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>π</mi> </mrow> <mi>λ</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>±</mo> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(\theta )=I_{0}\,\operatorname {sinc} ^{2}\left[{\frac {d\pi }{\lambda }}(\sin \theta \pm \sin \theta _{\text{i}})\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e5854942c75d71ac603f3275ce5b18866ed637f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.808ex; height:6.176ex;" alt="{\displaystyle I(\theta )=I_{0}\,\operatorname {sinc} ^{2}\left[{\frac {d\pi }{\lambda }}(\sin \theta \pm \sin \theta _{\text{i}})\right]}"></span> </p><p> The choice of plus/minus sign depends on the definition of the incident angle <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{\text{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{\text{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7f1cbab9315aa0f5290e429d9b2366f560dc39f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.78ex; height:2.509ex;" alt="{\displaystyle \theta _{\text{i}}}"></span>.</span></p><figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Diffraction2vs5.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Diffraction2vs5.jpg/220px-Diffraction2vs5.jpg" decoding="async" width="220" height="99" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/4/46/Diffraction2vs5.jpg 1.5x" data-file-width="259" data-file-height="116" /></a><figcaption>2-slit (top) and 5-slit diffraction of red laser light</figcaption></figure> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Diffraction-red_laser-diffraction_grating_PNr%C2%B00126.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Diffraction-red_laser-diffraction_grating_PNr%C2%B00126.jpg/220px-Diffraction-red_laser-diffraction_grating_PNr%C2%B00126.jpg" decoding="async" width="220" height="131" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Diffraction-red_laser-diffraction_grating_PNr%C2%B00126.jpg/330px-Diffraction-red_laser-diffraction_grating_PNr%C2%B00126.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Diffraction-red_laser-diffraction_grating_PNr%C2%B00126.jpg/440px-Diffraction-red_laser-diffraction_grating_PNr%C2%B00126.jpg 2x" data-file-width="3258" data-file-height="1939" /></a><figcaption>Diffraction of a red laser using a diffraction grating</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Diffraction_150_slits.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Diffraction_150_slits.jpg/220px-Diffraction_150_slits.jpg" decoding="async" width="220" height="86" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Diffraction_150_slits.jpg/330px-Diffraction_150_slits.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Diffraction_150_slits.jpg/440px-Diffraction_150_slits.jpg 2x" data-file-width="3296" data-file-height="1290" /></a><figcaption>A diffraction pattern of a 633 nm laser through a grid of 150 slits</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Diffraction_grating">Diffraction grating</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffraction&action=edit&section=5" title="Edit section: Diffraction grating"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Diffraction_grating" title="Diffraction grating">Diffraction grating</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><span><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Diffraction_grating_demo.webm/220px--Diffraction_grating_demo.webm.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" height="124" data-durationhint="46" data-mwtitle="Diffraction_grating_demo.webm" data-mwprovider="wikimediacommons" resource="/wiki/File:Diffraction_grating_demo.webm"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/63/Diffraction_grating_demo.webm/Diffraction_grating_demo.webm.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="854" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/63/Diffraction_grating_demo.webm/Diffraction_grating_demo.webm.720p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="720p.vp9.webm" data-width="1280" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/6/63/Diffraction_grating_demo.webm" type="video/webm; codecs="vp9, opus"" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/63/Diffraction_grating_demo.webm/Diffraction_grating_demo.webm.1080p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="1080p.vp9.webm" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/63/Diffraction_grating_demo.webm/Diffraction_grating_demo.webm.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="256" data-height="144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/63/Diffraction_grating_demo.webm/Diffraction_grating_demo.webm.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="426" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/63/Diffraction_grating_demo.webm/Diffraction_grating_demo.webm.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="640" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/63/Diffraction_grating_demo.webm/Diffraction_grating_demo.webm.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="640" data-height="360" /><track src="https://commons.wikimedia.org/w/api.php?action=timedtext&title=File%3ADiffraction_grating_demo.webm&lang=en&trackformat=vtt&origin=%2A" kind="subtitles" type="text/vtt" srclang="en" label="English ‪(en)‬" data-dir="ltr" /></video></span><figcaption>Diffraction grating</figcaption></figure> <p>A diffraction grating is an optical component with a regular pattern. The form of the light diffracted by a grating depends on the structure of the elements and the number of elements present, but all gratings have intensity maxima at angles <i>θ</i><sub><i>m</i></sub> which are given by the grating equation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\left(\sin {\theta _{m}}\pm \sin {\theta _{i}}\right)=m\lambda ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mo>±</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>m</mi> <mi>λ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\left(\sin {\theta _{m}}\pm \sin {\theta _{i}}\right)=m\lambda ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4cd0cf80f5ef683428dcfd37e6f277f380daa8f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.535ex; height:2.843ex;" alt="{\displaystyle d\left(\sin {\theta _{m}}\pm \sin {\theta _{i}}\right)=m\lambda ,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/302b19204ed378e99ff4575341a67eebdbe5a555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.89ex; height:2.509ex;" alt="{\displaystyle \theta _{i}}"></span> is the angle at which the light is incident, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> is the separation of grating elements, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> is an integer which can be positive or negative. </p><p>The light diffracted by a grating is found by summing the light diffracted from each of the elements, and is essentially a <a href="/wiki/Convolution" title="Convolution">convolution</a> of diffraction and interference patterns. </p><p>The figure shows the light diffracted by 2-element and 5-element gratings where the grating spacings are the same; it can be seen that the maxima are in the same position, but the detailed structures of the intensities are different. </p> <div class="mw-heading mw-heading3"><h3 id="Circular_aperture">Circular aperture</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffraction&action=edit&section=6" title="Edit section: Circular aperture"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Airy_disk" title="Airy disk">Airy disk</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Airy-pattern.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Airy-pattern.svg/220px-Airy-pattern.svg.png" decoding="async" width="220" height="163" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Airy-pattern.svg/330px-Airy-pattern.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/14/Airy-pattern.svg/440px-Airy-pattern.svg.png 2x" data-file-width="283" data-file-height="210" /></a><figcaption>A computer-generated image of an <b>Airy disk</b></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Fresnel_to_Fraunhofer_transition.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Fresnel_to_Fraunhofer_transition.gif/220px-Fresnel_to_Fraunhofer_transition.gif" decoding="async" width="220" height="227" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/a/a3/Fresnel_to_Fraunhofer_transition.gif 1.5x" data-file-width="301" data-file-height="310" /></a><figcaption>Diffraction pattern from a circular aperture at various distances</figcaption></figure> <p>The far-field diffraction of a plane wave incident on a circular aperture is often referred to as the <a href="/wiki/Airy_disk" title="Airy disk">Airy disk</a>. The <a href="/wiki/Airy_disk#Mathematical_details" title="Airy disk">variation</a> in intensity with angle is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(\theta )=I_{0}\left({\frac {2J_{1}(ka\sin \theta )}{ka\sin \theta }}\right)^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>k</mi> <mi>a</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>k</mi> <mi>a</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(\theta )=I_{0}\left({\frac {2J_{1}(ka\sin \theta )}{ka\sin \theta }}\right)^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/318e7761eedb211429406fd8ed3233620f23640e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.683ex; height:6.676ex;" alt="{\displaystyle I(\theta )=I_{0}\left({\frac {2J_{1}(ka\sin \theta )}{ka\sin \theta }}\right)^{2},}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> is the radius of the circular aperture, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> is equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi /\lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi /\lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c5bbdc15b292c06917367b94a551ab787fa9b1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.012ex; height:2.843ex;" alt="{\displaystyle 2\pi /\lambda }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/260ffe7da7c858cf114ad89a6c794944ea4e760f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.344ex; height:2.509ex;" alt="{\displaystyle J_{1}}"></span> is a <a href="/wiki/Bessel_function" title="Bessel function">Bessel function</a>. The smaller the aperture, the larger the spot size at a given distance, and the greater the divergence of the diffracted beams. </p> <div class="mw-heading mw-heading3"><h3 id="General_aperture">General aperture</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffraction&action=edit&section=7" title="Edit section: General aperture"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The wave that emerges from a point source has amplitude <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> at location <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"></span> that is given by the solution of the <a href="/wiki/Frequency_domain" title="Frequency domain">frequency domain</a> <a href="/wiki/Wave_equation" title="Wave equation">wave equation</a> for a point source (the <a href="/wiki/Helmholtz_equation" title="Helmholtz equation">Helmholtz equation</a>), <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla ^{2}\psi +k^{2}\psi =\delta (\mathbf {r} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ψ</mi> <mo>+</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ψ</mi> <mo>=</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}\psi +k^{2}\psi =\delta (\mathbf {r} ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d18f7118d163e259a431dee8d600dbc6ddd3cf8b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.828ex; height:3.176ex;" alt="{\displaystyle \nabla ^{2}\psi +k^{2}\psi =\delta (\mathbf {r} ),}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (\mathbf {r} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5c0d65c972b3ee05708309bd68570a928014ed6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.96ex; height:2.843ex;" alt="{\displaystyle \delta (\mathbf {r} )}"></span> is the 3-dimensional delta function. The delta function has only radial dependence, so the <a href="/wiki/Laplace_operator" title="Laplace operator">Laplace operator</a> (a.k.a. scalar Laplacian) in the <a href="/wiki/Spherical_coordinate_system" title="Spherical coordinate system">spherical coordinate system</a> simplifies to <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla ^{2}\psi ={\frac {1}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}(r\psi ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ψ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mi>ψ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}\psi ={\frac {1}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}(r\psi ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36ca8b068b12281e5b085f24858ee1a89289c158" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:18.875ex; height:6.009ex;" alt="{\displaystyle \nabla ^{2}\psi ={\frac {1}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}(r\psi ).}"></span> </p><p>(See <a href="/wiki/Del_in_cylindrical_and_spherical_coordinates" title="Del in cylindrical and spherical coordinates">del in cylindrical and spherical coordinates</a>.) By direct substitution, the solution to this equation can be readily shown to be the scalar <a href="/wiki/Green%27s_function" title="Green's function">Green's function</a>, which in the <a href="/wiki/Spherical_coordinate_system" title="Spherical coordinate system">spherical coordinate system</a> (and using the physics time convention <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-i\omega t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-i\omega t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c4bab37784e0c6bedf7e592b763d6aa59266c92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.778ex; height:2.676ex;" alt="{\displaystyle e^{-i\omega t}}"></span>) is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (r)={\frac {e^{ikr}}{4\pi r}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>r</mi> </mrow> </msup> <mrow> <mn>4</mn> <mi>π</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (r)={\frac {e^{ikr}}{4\pi r}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32c2df56924f9a331f0318fc96abeb05a3f1e5ed" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.496ex; height:5.676ex;" alt="{\displaystyle \psi (r)={\frac {e^{ikr}}{4\pi r}}.}"></span> </p><p>This solution assumes that the delta function source is located at the origin. If the source is located at an arbitrary source point, denoted by the vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} '}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1317b633a9366fab48e4b85ddec1cd1c0a8c31b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.509ex;" alt="{\displaystyle \mathbf {r} '}"></span> and the field point is located at the point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"></span>, then we may represent the scalar <a href="/wiki/Green%27s_function" title="Green's function">Green's function</a> (for arbitrary source location) as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\mathbf {r} |\mathbf {r} ')={\frac {e^{ik|\mathbf {r} -\mathbf {r} '|}}{4\pi |\mathbf {r} -\mathbf {r} '|}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> <mrow> <mn>4</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (\mathbf {r} |\mathbf {r} ')={\frac {e^{ik|\mathbf {r} -\mathbf {r} '|}}{4\pi |\mathbf {r} -\mathbf {r} '|}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/298e884d6c928adb6621db677d3a7f66d398ddea" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.957ex; height:6.676ex;" alt="{\displaystyle \psi (\mathbf {r} |\mathbf {r} ')={\frac {e^{ik|\mathbf {r} -\mathbf {r} '|}}{4\pi |\mathbf {r} -\mathbf {r} '|}}.}"></span> </p><p>Therefore, if an electric field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\mathrm {inc} }(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {inc} }(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c7adb32928a3ac9ed31b4e64e57cf42abb6b990" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.377ex; height:2.843ex;" alt="{\displaystyle E_{\mathrm {inc} }(x,y)}"></span> is incident on the aperture, the field produced by this aperture distribution is given by the <a href="/wiki/Surface_integral" title="Surface integral">surface integral</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi (r)\propto \iint \limits _{\mathrm {aperture} }\!\!E_{\mathrm {inc} }(x',y')~{\frac {e^{ik|\mathbf {r} -\mathbf {r} '|}}{4\pi |\mathbf {r} -\mathbf {r} '|}}\,dx'\,dy',}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>∝</mo> <munder> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </munder> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> <mrow> <mn>4</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (r)\propto \iint \limits _{\mathrm {aperture} }\!\!E_{\mathrm {inc} }(x',y')~{\frac {e^{ik|\mathbf {r} -\mathbf {r} '|}}{4\pi |\mathbf {r} -\mathbf {r} '|}}\,dx'\,dy',}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0df80f26bc76c6149fc3ecd175550ed3ab624cbf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:42.229ex; height:8.176ex;" alt="{\displaystyle \Psi (r)\propto \iint \limits _{\mathrm {aperture} }\!\!E_{\mathrm {inc} }(x',y')~{\frac {e^{ik|\mathbf {r} -\mathbf {r} '|}}{4\pi |\mathbf {r} -\mathbf {r} '|}}\,dx'\,dy',}"></span> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Fraunhofer.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Fraunhofer.svg/310px-Fraunhofer.svg.png" decoding="async" width="310" height="178" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Fraunhofer.svg/465px-Fraunhofer.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Fraunhofer.svg/620px-Fraunhofer.svg.png 2x" data-file-width="805" data-file-height="463" /></a><figcaption>On the calculation of Fraunhofer region fields</figcaption></figure> <p>where the source point in the aperture is given by the vector <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} '=x'\mathbf {\hat {x}} +y'\mathbf {\hat {y}} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo>=</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">x</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">y</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} '=x'\mathbf {\hat {x}} +y'\mathbf {\hat {y}} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1352fb48fd3849697a46368c48f14b21a334539" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.054ex; height:2.843ex;" alt="{\displaystyle \mathbf {r} '=x'\mathbf {\hat {x}} +y'\mathbf {\hat {y}} .}"></span> </p><p>In the far field, wherein the parallel rays approximation can be employed, the Green's function, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\mathbf {r} |\mathbf {r} ')={\frac {e^{ik|\mathbf {r} -\mathbf {r} '|}}{4\pi |\mathbf {r} -\mathbf {r} '|}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> <mrow> <mn>4</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (\mathbf {r} |\mathbf {r} ')={\frac {e^{ik|\mathbf {r} -\mathbf {r} '|}}{4\pi |\mathbf {r} -\mathbf {r} '|}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27ff2e67ecbe945e5f0c2cb7a8d2cdeb34c12479" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.957ex; height:6.676ex;" alt="{\displaystyle \psi (\mathbf {r} |\mathbf {r} ')={\frac {e^{ik|\mathbf {r} -\mathbf {r} '|}}{4\pi |\mathbf {r} -\mathbf {r} '|}},}"></span> simplifies to <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (\mathbf {r} |\mathbf {r} ')={\frac {e^{ikr}}{4\pi r}}e^{-ik(\mathbf {r} '\cdot \mathbf {\hat {r}} )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>r</mi> </mrow> </msup> <mrow> <mn>4</mn> <mi>π</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>k</mi> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">r</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (\mathbf {r} |\mathbf {r} ')={\frac {e^{ikr}}{4\pi r}}e^{-ik(\mathbf {r} '\cdot \mathbf {\hat {r}} )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3109378c66517e8f969b7d4039a8ebcfc88af8fd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.347ex; height:5.676ex;" alt="{\displaystyle \psi (\mathbf {r} |\mathbf {r} ')={\frac {e^{ikr}}{4\pi r}}e^{-ik(\mathbf {r} '\cdot \mathbf {\hat {r}} )}}"></span> as can be seen in the adjacent figure. </p><p>The expression for the far-zone (Fraunhofer region) field becomes <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi (r)\propto {\frac {e^{ikr}}{4\pi r}}\iint \limits _{\mathrm {aperture} }\!\!E_{\mathrm {inc} }(x',y')e^{-ik(\mathbf {r} '\cdot \mathbf {\hat {r}} )}\,dx'\,dy'.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>∝</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>r</mi> </mrow> </msup> <mrow> <mn>4</mn> <mi>π</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <munder> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </munder> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>k</mi> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">r</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (r)\propto {\frac {e^{ikr}}{4\pi r}}\iint \limits _{\mathrm {aperture} }\!\!E_{\mathrm {inc} }(x',y')e^{-ik(\mathbf {r} '\cdot \mathbf {\hat {r}} )}\,dx'\,dy'.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bc391b1b91d6ce96e1074c2259f131a95f1d2ba" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:44.072ex; height:8.009ex;" alt="{\displaystyle \Psi (r)\propto {\frac {e^{ikr}}{4\pi r}}\iint \limits _{\mathrm {aperture} }\!\!E_{\mathrm {inc} }(x',y')e^{-ik(\mathbf {r} '\cdot \mathbf {\hat {r}} )}\,dx'\,dy'.}"></span> </p><p>Now, since <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} '=x'\mathbf {\hat {x}} +y'\mathbf {\hat {y}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo>=</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">x</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">y</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} '=x'\mathbf {\hat {x}} +y'\mathbf {\hat {y}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5920361f62c5aa68d1326b4810ba37f646bab43f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.407ex; height:2.843ex;" alt="{\displaystyle \mathbf {r} '=x'\mathbf {\hat {x}} +y'\mathbf {\hat {y}} }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {r}} =\sin \theta \cos \phi \mathbf {\hat {x}} +\sin \theta ~\sin \phi ~\mathbf {\hat {y}} +\cos \theta \mathbf {\hat {z}} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">r</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">x</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">y</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">z</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {r}} =\sin \theta \cos \phi \mathbf {\hat {x}} +\sin \theta ~\sin \phi ~\mathbf {\hat {y}} +\cos \theta \mathbf {\hat {z}} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67a9f65ffd57acf718f256230af16e176880615a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:39.624ex; height:2.676ex;" alt="{\displaystyle \mathbf {\hat {r}} =\sin \theta \cos \phi \mathbf {\hat {x}} +\sin \theta ~\sin \phi ~\mathbf {\hat {y}} +\cos \theta \mathbf {\hat {z}} ,}"></span> the expression for the Fraunhofer region field from a planar aperture now becomes <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi (r)\propto {\frac {e^{ikr}}{4\pi r}}\iint \limits _{\mathrm {aperture} }\!\!E_{\mathrm {inc} }(x',y')e^{-ik\sin \theta (\cos \phi x'+\sin \phi y')}\,dx'\,dy'.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>∝</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>r</mi> </mrow> </msup> <mrow> <mn>4</mn> <mi>π</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <munder> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </munder> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>k</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (r)\propto {\frac {e^{ikr}}{4\pi r}}\iint \limits _{\mathrm {aperture} }\!\!E_{\mathrm {inc} }(x',y')e^{-ik\sin \theta (\cos \phi x'+\sin \phi y')}\,dx'\,dy'.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fc6a943d922a20506dff10ff61de81a98bb6ede" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:55.978ex; height:8.009ex;" alt="{\displaystyle \Psi (r)\propto {\frac {e^{ikr}}{4\pi r}}\iint \limits _{\mathrm {aperture} }\!\!E_{\mathrm {inc} }(x',y')e^{-ik\sin \theta (\cos \phi x'+\sin \phi y')}\,dx'\,dy'.}"></span> </p><p>Letting <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{x}=k\sin \theta \cos \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mi>k</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{x}=k\sin \theta \cos \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45022c7b465259acb6ee917117e7d42a67cfdcb6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.684ex; height:2.509ex;" alt="{\displaystyle k_{x}=k\sin \theta \cos \phi }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{y}=k\sin \theta \sin \phi \,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mi>k</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{y}=k\sin \theta \sin \phi \,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7422bd5e8d8b918f2a8708d5e140a314f6737b6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.34ex; height:2.843ex;" alt="{\displaystyle k_{y}=k\sin \theta \sin \phi \,,}"></span> the Fraunhofer region field of the planar aperture assumes the form of a <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi (r)\propto {\frac {e^{ikr}}{4\pi r}}\iint \limits _{\mathrm {aperture} }\!\!E_{\mathrm {inc} }(x',y')e^{-i(k_{x}x'+k_{y}y')}\,dx'\,dy',}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>∝</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>r</mi> </mrow> </msup> <mrow> <mn>4</mn> <mi>π</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <munder> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </munder> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (r)\propto {\frac {e^{ikr}}{4\pi r}}\iint \limits _{\mathrm {aperture} }\!\!E_{\mathrm {inc} }(x',y')e^{-i(k_{x}x'+k_{y}y')}\,dx'\,dy',}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddaca41027387a5ea0a9022c87e9d9763477ced0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:48.073ex; height:8.009ex;" alt="{\displaystyle \Psi (r)\propto {\frac {e^{ikr}}{4\pi r}}\iint \limits _{\mathrm {aperture} }\!\!E_{\mathrm {inc} }(x',y')e^{-i(k_{x}x'+k_{y}y')}\,dx'\,dy',}"></span> </p><p>In the far-field / Fraunhofer region, this becomes the spatial <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> of the aperture distribution. Huygens' principle when applied to an aperture simply says that the <a href="/wiki/Far-field_diffraction_pattern" class="mw-redirect" title="Far-field diffraction pattern">far-field diffraction pattern</a> is the spatial Fourier transform of the aperture shape, and this is a direct by-product of using the parallel-rays approximation, which is identical to doing a plane wave decomposition of the aperture plane fields (see <a href="/wiki/Fourier_optics" title="Fourier optics">Fourier optics</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Propagation_of_a_laser_beam">Propagation of a laser beam</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffraction&action=edit&section=8" title="Edit section: Propagation of a laser beam"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The way in which the beam profile of a <a href="/wiki/Laser" title="Laser">laser beam</a> changes as it propagates is determined by diffraction. When the entire emitted beam has a planar, spatially <a href="/wiki/Coherence_(physics)" title="Coherence (physics)">coherent</a> wave front, it approximates <a href="/wiki/Gaussian_beam" title="Gaussian beam">Gaussian beam</a> profile and has the lowest divergence for a given diameter. The smaller the output beam, the quicker it diverges. It is possible to reduce the divergence of a laser beam by first expanding it with one <a href="/wiki/Convex_lens" class="mw-redirect" title="Convex lens">convex lens</a>, and then collimating it with a second convex lens whose focal point is coincident with that of the first lens. The resulting beam has a larger diameter, and hence a lower divergence. Divergence of a laser beam may be reduced below the diffraction of a Gaussian beam or even reversed to convergence if the refractive index of the propagation media increases with the light intensity.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> This may result in a <a href="/wiki/Self-focusing" title="Self-focusing">self-focusing</a> effect. </p><p>When the wave front of the emitted beam has perturbations, only the transverse coherence length (where the wave front perturbation is less than 1/4 of the wavelength) should be considered as a Gaussian beam diameter when determining the divergence of the laser beam. If the transverse coherence length in the vertical direction is higher than in horizontal, the laser beam divergence will be lower in the vertical direction than in the horizontal. </p> <div class="mw-heading mw-heading3"><h3 id="Diffraction-limited_imaging">Diffraction-limited imaging</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffraction&action=edit&section=9" title="Edit section: Diffraction-limited imaging"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Diffraction-limited_system" title="Diffraction-limited system">Diffraction-limited system</a></div> <figure typeof="mw:File/Frame"><a href="/wiki/File:Zboo_lucky_image_1pc.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/b6/Zboo_lucky_image_1pc.png" decoding="async" width="224" height="183" class="mw-file-element" data-file-width="224" data-file-height="183" /></a><figcaption>The Airy disk around each of the stars from the 2.56 m telescope aperture can be seen in this <i><a href="/wiki/Lucky_imaging" title="Lucky imaging">lucky image</a></i> of the <a href="/wiki/Binary_star" title="Binary star">binary star</a> <a href="/wiki/Zeta_Bo%C3%B6tis" title="Zeta Boötis">zeta Boötis</a>.</figcaption></figure> <p>The ability of an imaging system to resolve detail is ultimately limited by <a href="/wiki/Diffraction-limited" class="mw-redirect" title="Diffraction-limited">diffraction</a>. This is because a plane wave incident on a circular lens or mirror is diffracted as described above. The light is not focused to a point but forms an Airy disk having a central spot in the focal plane whose radius (as measured to the first null) is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x=1.22\lambda N,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>=</mo> <mn>1.22</mn> <mi>λ</mi> <mi>N</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x=1.22\lambda N,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0bf219ac8262d284d284e388c0bc8a1482e548a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.564ex; height:2.509ex;" alt="{\displaystyle \Delta x=1.22\lambda N,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> is the wavelength of the light and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> is the <a href="/wiki/F-number" title="F-number">f-number</a> (focal length <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> divided by aperture diameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span>) of the imaging optics; this is strictly accurate for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\gg 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>≫</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N\gg 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7067e3506c2342958dfbe0058ae7563915b852c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.84ex; height:2.176ex;" alt="{\displaystyle N\gg 1}"></span> (<a href="/wiki/Paraxial" class="mw-redirect" title="Paraxial">paraxial</a> case). In object space, the corresponding <a href="/wiki/Angular_resolution" title="Angular resolution">angular resolution</a> is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta \approx \sin \theta =1.22{\frac {\lambda }{D}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>≈</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mn>1.22</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mi>D</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta \approx \sin \theta =1.22{\frac {\lambda }{D}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b38e5c3d13d771819c80e99197ad9e7829a3422" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.162ex; height:5.343ex;" alt="{\displaystyle \theta \approx \sin \theta =1.22{\frac {\lambda }{D}},}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> is the diameter of the <a href="/wiki/Entrance_pupil" title="Entrance pupil">entrance pupil</a> of the imaging lens (e.g., of a telescope's main mirror). </p><p>Two point sources will each produce an Airy pattern – see the photo of a binary star. As the point sources move closer together, the patterns will start to overlap, and ultimately they will merge to form a single pattern, in which case the two point sources cannot be resolved in the image. The <a href="/wiki/Angular_resolution#The_Rayleigh_criterion" title="Angular resolution">Rayleigh criterion</a> specifies that two point sources are considered "resolved" if the separation of the two images is at least the radius of the Airy disk, i.e. if the first minimum of one coincides with the maximum of the other. </p><p>Thus, the larger the aperture of the lens compared to the wavelength, the finer the resolution of an imaging system. This is one reason astronomical telescopes require large objectives, and why <a href="/wiki/Objective_(optics)#Microscope" title="Objective (optics)">microscope objectives</a> require a large <a href="/wiki/Numerical_aperture" title="Numerical aperture">numerical aperture</a> (large aperture diameter compared to working distance) in order to obtain the highest possible resolution. </p> <div class="mw-heading mw-heading3"><h3 id="Speckle_patterns">Speckle patterns</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffraction&action=edit&section=10" title="Edit section: Speckle patterns"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Speckle_pattern" class="mw-redirect" title="Speckle pattern">Speckle pattern</a></div> <p>The <a href="/wiki/Speckle_pattern" class="mw-redirect" title="Speckle pattern">speckle pattern</a> seen when using a <a href="/wiki/Laser_pointer" title="Laser pointer">laser pointer</a> is another diffraction phenomenon. It is a result of the superposition of many waves with different phases, which are produced when a laser beam illuminates a rough surface. They add together to give a resultant wave whose amplitude, and therefore intensity, varies randomly. </p> <div class="mw-heading mw-heading3"><h3 id="Babinet's_principle"><span id="Babinet.27s_principle"></span>Babinet's principle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffraction&action=edit&section=11" title="Edit section: Babinet's principle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Babinet%27s_principle" title="Babinet's principle">Babinet's principle</a></div> <p><a href="/wiki/Babinet%27s_principle" title="Babinet's principle">Babinet's principle</a> is a useful theorem stating that the diffraction pattern from an opaque body is identical to that from a hole of the same size and shape, but with differing intensities. This means that the interference conditions of a single obstruction would be the same as that of a single slit. </p> <div class="mw-heading mw-heading3"><h3 id=""Knife_edge""><span id=".22Knife_edge.22"></span>"Knife edge" <span class="anchor" id="Knife_edge"></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffraction&action=edit&section=12" title="Edit section: "Knife edge""><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>knife-edge effect</b> or <b>knife-edge diffraction</b> is a truncation of a portion of the incident <a href="/wiki/Radiation" title="Radiation">radiation</a> that strikes a sharp well-defined obstacle, such as a mountain range or the wall of a building. The knife-edge effect is explained by the <a href="/wiki/Huygens%E2%80%93Fresnel_principle" title="Huygens–Fresnel principle">Huygens–Fresnel principle</a>, which states that a well-defined obstruction to an electromagnetic wave acts as a secondary source, and creates a new <a href="/wiki/Wavefront" title="Wavefront">wavefront</a>. This new wavefront propagates into the geometric shadow area of the obstacle. </p><p>Knife-edge diffraction is an outgrowth of the "<a href="/wiki/Half-plane" class="mw-redirect" title="Half-plane">half-plane</a> problem", originally solved by <a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Arnold Sommerfeld</a> using a plane wave spectrum formulation. A generalization of the half-plane problem is the "wedge problem", solvable as a boundary value problem in cylindrical coordinates. The solution in cylindrical coordinates was then extended to the optical regime by <a href="/wiki/Joseph_B._Keller" class="mw-redirect" title="Joseph B. Keller">Joseph B. Keller</a>, who introduced the notion of diffraction coefficients through his <a href="/wiki/Geometrical_theory_of_diffraction" class="mw-redirect" title="Geometrical theory of diffraction">geometrical theory of diffraction</a> (GTD). In 1974, Prabhakar Pathak and <a href="/wiki/Robert_Kouyoumjian" title="Robert Kouyoumjian">Robert Kouyoumjian</a> extended the (singular) Keller coefficients via the <a href="/wiki/Uniform_theory_of_diffraction" title="Uniform theory of diffraction">uniform theory of diffraction</a> (UTD).<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <ul class="gallery mw-gallery-nolines"> <li class="gallerybox" style="width: 305px"> <div class="thumb" style="width: 300px;"><span typeof="mw:File"><a href="/wiki/File:Diffraction_sharp_edge.gif" class="mw-file-description" title="Diffraction on a sharp metallic edge"><img alt="Diffraction on a sharp metallic edge" src="//upload.wikimedia.org/wikipedia/commons/4/4b/Diffraction_sharp_edge.gif" decoding="async" width="168" height="238" class="mw-file-element" data-file-width="168" data-file-height="238" /></a></span></div> <div class="gallerytext">Diffraction on a sharp metallic edge</div> </li> <li class="gallerybox" style="width: 305px"> <div class="thumb" style="width: 300px;"><span typeof="mw:File"><a href="/wiki/File:Diffraction_softest_edge.gif" class="mw-file-description" title="Diffraction on a soft aperture, with a gradient of conductivity over the image width"><img alt="Diffraction on a soft aperture, with a gradient of conductivity over the image width" src="//upload.wikimedia.org/wikipedia/commons/5/53/Diffraction_softest_edge.gif" decoding="async" width="168" height="238" class="mw-file-element" data-file-width="168" data-file-height="238" /></a></span></div> <div class="gallerytext">Diffraction on a soft aperture, with a gradient of conductivity over the image width</div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="Patterns">Patterns</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffraction&action=edit&section=13" title="Edit section: Patterns"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Diffraction_on_elliptic_aperture_with_fft.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Diffraction_on_elliptic_aperture_with_fft.png/220px-Diffraction_on_elliptic_aperture_with_fft.png" decoding="async" width="220" height="328" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Diffraction_on_elliptic_aperture_with_fft.png/330px-Diffraction_on_elliptic_aperture_with_fft.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Diffraction_on_elliptic_aperture_with_fft.png/440px-Diffraction_on_elliptic_aperture_with_fft.png 2x" data-file-width="556" data-file-height="828" /></a><figcaption>The upper half of this image shows a diffraction pattern of He-Ne laser beam on an elliptic aperture. The lower half is its 2D Fourier transform approximately reconstructing the shape of the aperture.</figcaption></figure> <p>Several qualitative observations can be made of diffraction in general: </p> <ul><li>The angular spacing of the features in the diffraction pattern is inversely proportional to the dimensions of the object causing the diffraction. In other words: The smaller the diffracting object, the 'wider' the resulting diffraction pattern, and vice versa. (More precisely, this is true of the <a href="/wiki/Sine" class="mw-redirect" title="Sine">sines</a> of the angles.)</li> <li>The diffraction angles are invariant under scaling; that is, they depend only on the ratio of the wavelength to the size of the diffracting object.</li> <li>When the diffracting object has a periodic structure, for example in a diffraction grating, the features generally become sharper. The third figure, for example, shows a comparison of a <a href="/wiki/Double-slit_experiment" title="Double-slit experiment">double-slit</a> pattern with a pattern formed by five slits, both sets of slits having the same spacing, between the center of one slit and the next.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Matter_wave_diffraction">Matter wave diffraction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffraction&action=edit&section=14" title="Edit section: Matter wave diffraction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Matter_wave" title="Matter wave">Matter wave</a>, <a href="/wiki/Neutron_diffraction" title="Neutron diffraction">Neutron diffraction</a>, and <a href="/wiki/Electron_diffraction" title="Electron diffraction">Electron diffraction</a></div> <p>According to quantum theory every particle exhibits wave properties and can therefore diffract. Diffraction of electrons and neutrons is one of the powerful arguments in favor of quantum mechanics. The wavelength associated with a particle is the <a href="/wiki/De_Broglie_wavelength" class="mw-redirect" title="De Broglie wavelength">de Broglie wavelength</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ={\frac {h}{p}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mi>p</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ={\frac {h}{p}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57f18cc0bc210e3e91b4a9a3f4e868717978ec6c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:7.663ex; height:5.843ex;" alt="{\displaystyle \lambda ={\frac {h}{p}}\,,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> is the <a href="/wiki/Planck_constant" title="Planck constant">Planck constant</a> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> is the <a href="/wiki/Momentum" title="Momentum">momentum</a> of the particle (mass × velocity for slow-moving particles). For example, a sodium atom traveling at about 300 m/s would have a de Broglie wavelength of about 50 picometres. </p><p>Diffraction of <a href="/wiki/Matter_wave" title="Matter wave">matter waves</a> has been observed for small particles, like electrons, neutrons, atoms, and even large molecules. The short wavelength of these matter waves makes them ideally suited to study the atomic crystal structure of solids, small molecules and proteins. </p> <div class="mw-heading mw-heading2"><h2 id="Bragg_diffraction">Bragg diffraction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffraction&action=edit&section=15" title="Edit section: Bragg diffraction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Bragg_diffraction" class="mw-redirect" title="Bragg diffraction">Bragg diffraction</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:X-ray_diffraction_pattern_3clpro.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/X-ray_diffraction_pattern_3clpro.jpg/220px-X-ray_diffraction_pattern_3clpro.jpg" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/X-ray_diffraction_pattern_3clpro.jpg/330px-X-ray_diffraction_pattern_3clpro.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7d/X-ray_diffraction_pattern_3clpro.jpg/440px-X-ray_diffraction_pattern_3clpro.jpg 2x" data-file-width="678" data-file-height="677" /></a><figcaption>Following <a href="/wiki/Bragg%27s_law" title="Bragg's law">Bragg's law</a>, each dot (or <i>reflection</i>) in this diffraction pattern forms from the constructive interference of X-rays passing through a crystal. The data can be used to determine the crystal's atomic structure.</figcaption></figure> <p>Diffraction from a large three-dimensional periodic structure such as many thousands of atoms in a crystal is called <a href="/wiki/Bragg_diffraction" class="mw-redirect" title="Bragg diffraction">Bragg diffraction</a>. It is similar to what occurs when waves are scattered from a <a href="/wiki/Diffraction_grating" title="Diffraction grating">diffraction grating</a>. Bragg diffraction is a consequence of interference between waves reflecting from many different crystal planes. The condition of constructive interference is given by <i>Bragg's law</i>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\lambda =2d\sin \theta ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mi>λ</mi> <mo>=</mo> <mn>2</mn> <mi>d</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\lambda =2d\sin \theta ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c918a1965c82cb3b240eccc22d814d1c2f5d0175" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.24ex; height:2.509ex;" alt="{\displaystyle m\lambda =2d\sin \theta ,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> is the wavelength, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> is the distance between crystal planes, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> is the angle of the diffracted wave, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> is an integer known as the <i>order</i> of the diffracted beam. </p><p>Bragg diffraction may be carried out using either electromagnetic radiation of very short wavelength like <a href="/wiki/X-ray_crystallography" title="X-ray crystallography">X-rays</a> or matter waves like <a href="/wiki/Neutron_diffraction" title="Neutron diffraction">neutrons</a> (and <a href="/wiki/Electron_diffraction" title="Electron diffraction">electrons</a>) whose wavelength is on the order of (or much smaller than) the atomic spacing.<sup id="cite_ref-JMC_23-0" class="reference"><a href="#cite_note-JMC-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> The pattern produced gives information of the separations of crystallographic planes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span>, allowing one to deduce the crystal structure. </p><p>For completeness, Bragg diffraction is a limit for a large number of atoms with X-rays or neutrons, and is rarely valid for <a href="/wiki/Electron_diffraction" title="Electron diffraction">electron diffraction</a> or with solid particles in the size range of less than 50 nanometers.<sup id="cite_ref-JMC_23-1" class="reference"><a href="#cite_note-JMC-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Coherence">Coherence</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffraction&action=edit&section=16" title="Edit section: Coherence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Coherence_(physics)" title="Coherence (physics)">Coherence (physics)</a></div> <p>The description of diffraction relies on the interference of waves emanating from the same source taking different paths to the same point on a screen. In this description, the difference in phase between waves that took different paths is only dependent on the effective path length. This does not take into account the fact that waves that arrive at the screen at the same time were emitted by the source at different times. The initial phase with which the source emits waves can change over time in an unpredictable way. This means that waves emitted by the source at times that are too far apart can no longer form a constant interference pattern since the relation between their phases is no longer time independent.<sup id="cite_ref-Halliday_24-0" class="reference"><a href="#cite_note-Halliday-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 919">: 919 </span></sup> </p><p>The length over which the phase in a beam of light is correlated is called the <a href="/wiki/Coherence_length" title="Coherence length">coherence length</a>. In order for interference to occur, the path length difference must be smaller than the coherence length. This is sometimes referred to as spectral coherence, as it is related to the presence of different frequency components in the wave. In the case of light emitted by an <a href="/wiki/Energy_level" title="Energy level">atomic transition</a>, the coherence length is related to the lifetime of the excited state from which the atom made its transition.<sup id="cite_ref-Fowles1975_25-0" class="reference"><a href="#cite_note-Fowles1975-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Pages: 71–74">: 71–74 </span></sup><sup id="cite_ref-Hecht2002_26-0" class="reference"><a href="#cite_note-Hecht2002-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Pages: 314–316">: 314–316 </span></sup> </p><p>If waves are emitted from an extended source, this can lead to incoherence in the transversal direction. When looking at a cross section of a beam of light, the length over which the phase is correlated is called the transverse coherence length. In the case of Young's double-slit experiment, this would mean that if the transverse coherence length is smaller than the spacing between the two slits, the resulting pattern on a screen would look like two single-slit diffraction patterns.<sup id="cite_ref-Fowles1975_25-1" class="reference"><a href="#cite_note-Fowles1975-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Pages: 74–79">: 74–79 </span></sup> </p><p>In the case of particles like electrons, neutrons, and atoms, the coherence length is related to the spatial extent of the wave function that describes the particle.<sup id="cite_ref-IchimiyaCohen2004_27-0" class="reference"><a href="#cite_note-IchimiyaCohen2004-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 107">: 107 </span></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffraction&action=edit&section=17" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Diffraction_before_destruction">Diffraction before destruction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffraction&action=edit&section=18" title="Edit section: Diffraction before destruction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A new way to image single biological particles has emerged since the 2010s, utilising the bright X-rays generated by X-ray <a href="/wiki/Free-electron_laser" title="Free-electron laser">free-electron lasers</a>. These femtosecond-duration pulses will allow for the (potential) imaging of single biological macromolecules. Due to these short pulses, radiation damage can be outrun, and diffraction patterns of single biological macromolecules will be able to be obtained.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffraction&action=edit&section=19" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 22em;"> <ul><li><a href="/wiki/Angle-sensitive_pixel" title="Angle-sensitive pixel">Angle-sensitive pixel</a></li> <li><a href="/wiki/Atmospheric_diffraction" title="Atmospheric diffraction">Atmospheric diffraction</a></li> <li><a href="/wiki/Brocken_spectre" title="Brocken spectre">Brocken spectre</a></li> <li><a href="/wiki/Cloud_iridescence" title="Cloud iridescence">Cloud iridescence</a></li> <li><a href="/wiki/Coherent_diffraction_imaging" title="Coherent diffraction imaging">Coherent diffraction imaging</a></li> <li><a href="/wiki/Diffraction_from_slits" title="Diffraction from slits">Diffraction from slits</a></li> <li><a href="/wiki/Diffraction_spike" title="Diffraction spike">Diffraction spike</a></li> <li><a href="/wiki/Diffraction_vs._interference" class="mw-redirect" title="Diffraction vs. interference">Diffraction vs. interference</a></li> <li><a href="/wiki/Diffractive_solar_sail" title="Diffractive solar sail">Diffractive solar sail</a></li> <li><a href="/wiki/Diffractometer" title="Diffractometer">Diffractometer</a></li> <li><a href="/wiki/Dynamical_theory_of_diffraction" title="Dynamical theory of diffraction">Dynamical theory of diffraction</a></li> <li><a href="/wiki/Electron_diffraction" title="Electron diffraction">Electron diffraction</a></li> <li><a href="/wiki/Fraunhofer_diffraction" title="Fraunhofer diffraction">Fraunhofer diffraction</a></li> <li><a href="/wiki/Fresnel_imager" title="Fresnel imager">Fresnel imager</a></li> <li><a href="/wiki/Fresnel_number" title="Fresnel number">Fresnel number</a></li> <li><a href="/wiki/Fresnel_zone" title="Fresnel zone">Fresnel zone</a></li> <li><a href="/wiki/Point_spread_function" title="Point spread function">Point spread function</a></li> <li><a href="/wiki/Powder_diffraction" title="Powder diffraction">Powder diffraction</a></li> <li><a href="/wiki/Quasioptics" title="Quasioptics">Quasioptics</a></li> <li><a href="/wiki/Refraction" title="Refraction">Refraction</a></li> <li><a href="/wiki/Reflection_(physics)" title="Reflection (physics)">Reflection</a></li> <li><a href="/wiki/Schaefer%E2%80%93Bergmann_diffraction" title="Schaefer–Bergmann diffraction">Schaefer–Bergmann diffraction</a></li> <li><a href="/wiki/Thinned-array_curse" title="Thinned-array curse">Thinned-array curse</a></li> <li><a href="/wiki/X-ray_diffraction" title="X-ray diffraction">X-ray diffraction</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffraction&action=edit&section=20" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Francesco Maria Grimaldi, <i>Physico mathesis de lumine, coloribus, et iride, aliisque annexis libri duo</i> (Bologna ("Bonomia"), Italy: Vittorio Bonati, 1665), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=FzYVAAAAQAAJ&pg=PA2">page 2</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161201153749/https://books.google.com/books?id=FzYVAAAAQAAJ&pg=PA2">Archived</a> 2016-12-01 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>: <blockquote><p><i>Original</i> : Nobis alius quartus modus illuxit, quem nunc proponimus, vocamusque; diffractionem, quia advertimus lumen aliquando diffringi, hoc est partes eius multiplici dissectione separatas per idem tamen medium in diversa ulterius procedere, eo modo, quem mox declarabimus.</p></blockquote> <blockquote><p><i>Translation</i> : It has illuminated for us another, fourth way, which we now make known and call "diffraction" [i.e., shattering], because we sometimes observe light break up; that is, that parts of the compound [i.e., the beam of light], separated by division, advance farther through the medium but in different [directions], as we will soon show.</p></blockquote></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Cajori, Florian <a rel="nofollow" class="external text" href="https://archive.org/details/ahistoryphysics00cajogoog/page/n102">"A History of Physics in its Elementary Branches, including the evolution of physical laboratories."</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161201075614/https://books.google.com/books?id=KZ4C-1CRtYQC&ots=c_YpkkbTpT&dq=Florian%20Cajori%20history%20of%20physics&pg=PA88">Archived</a> 2016-12-01 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> MacMillan Company, New York 1899</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Wireless Communications: Principles and Practice, Prentice Hall communications engineering and emerging technologies series, T. S. Rappaport, Prentice Hall, 2002 pg 126</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFSuryanarayanaNorton2013" class="citation book cs1">Suryanarayana, C.; Norton, M. Grant (29 June 2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RRfrBwAAQBAJ"><i>X-Ray Diffraction: A Practical Approach</i></a>. Springer Science & Business Media. p. 14. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4899-0148-4" title="Special:BookSources/978-1-4899-0148-4"><bdi>978-1-4899-0148-4</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">7 January</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=X-Ray+Diffraction%3A+A+Practical+Approach&rft.pages=14&rft.pub=Springer+Science+%26+Business+Media&rft.date=2013-06-29&rft.isbn=978-1-4899-0148-4&rft.aulast=Suryanarayana&rft.aufirst=C.&rft.au=Norton%2C+M.+Grant&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DRRfrBwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiffraction" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKokkotas2003" class="citation journal cs1">Kokkotas, Kostas D. (2003). "Gravitational Wave Physics". <i>Encyclopedia of Physical Science and Technology</i>: 67–85. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FB0-12-227410-5%2F00300-8">10.1016/B0-12-227410-5/00300-8</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780122274107" title="Special:BookSources/9780122274107"><bdi>9780122274107</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Encyclopedia+of+Physical+Science+and+Technology&rft.atitle=Gravitational+Wave+Physics&rft.pages=67-85&rft.date=2003&rft_id=info%3Adoi%2F10.1016%2FB0-12-227410-5%2F00300-8&rft.isbn=9780122274107&rft.aulast=Kokkotas&rft.aufirst=Kostas+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiffraction" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJuffmannMilicMüllneritschAsenbaum2012" class="citation journal cs1">Juffmann, Thomas; Milic, Adriana; Müllneritsch, Michael; Asenbaum, Peter; Tsukernik, Alexander; Tüxen, Jens; Mayor, Marcel; Cheshnovsky, Ori; Arndt, Markus (25 March 2012). "Real-time single-molecule imaging of quantum interference". <i>Nature Nanotechnology</i>. <b>7</b> (5): 297–300. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1402.1867">1402.1867</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2012NatNa...7..297J">2012NatNa...7..297J</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnnano.2012.34">10.1038/nnano.2012.34</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1748-3395">1748-3395</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/22447163">22447163</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:5918772">5918772</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature+Nanotechnology&rft.atitle=Real-time+single-molecule+imaging+of+quantum+interference&rft.volume=7&rft.issue=5&rft.pages=297-300&rft.date=2012-03-25&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A5918772%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2012NatNa...7..297J&rft_id=info%3Aarxiv%2F1402.1867&rft.issn=1748-3395&rft_id=info%3Adoi%2F10.1038%2Fnnano.2012.34&rft_id=info%3Apmid%2F22447163&rft.aulast=Juffmann&rft.aufirst=Thomas&rft.au=Milic%2C+Adriana&rft.au=M%C3%BCllneritsch%2C+Michael&rft.au=Asenbaum%2C+Peter&rft.au=Tsukernik%2C+Alexander&rft.au=T%C3%BCxen%2C+Jens&rft.au=Mayor%2C+Marcel&rft.au=Cheshnovsky%2C+Ori&rft.au=Arndt%2C+Markus&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiffraction" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKomechMerzon2019" class="citation cs2">Komech, Alexander; Merzon, Anatoli (2019), Komech, Alexander; Merzon, Anatoli (eds.), <a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-030-26699-8_2">"The Early Theory of Diffraction"</a>, <i>Stationary Diffraction by Wedges : Method of Automorphic Functions on Complex Characteristics</i>, Cham: Springer International Publishing, pp. 15–17, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-030-26699-8_2">10.1007/978-3-030-26699-8_2</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-030-26699-8" title="Special:BookSources/978-3-030-26699-8"><bdi>978-3-030-26699-8</bdi></a><span class="reference-accessdate">, retrieved <span class="nowrap">25 April</span> 2024</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Stationary+Diffraction+by+Wedges+%3A+Method+of+Automorphic+Functions+on+Complex+Characteristics&rft.atitle=The+Early+Theory+of+Diffraction&rft.pages=15-17&rft.date=2019&rft_id=info%3Adoi%2F10.1007%2F978-3-030-26699-8_2&rft.isbn=978-3-030-26699-8&rft.aulast=Komech&rft.aufirst=Alexander&rft.au=Merzon%2C+Anatoli&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%2F978-3-030-26699-8_2&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiffraction" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Francesco Maria Grimaldi, <i>Physico-mathesis de lumine, coloribus, et iride, aliisque adnexis …</i> [The physical mathematics of light, color, and the rainbow, and other things appended …] (Bologna ("Bonomia"), (Italy): Vittorio Bonati, 1665), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=FzYVAAAAQAAJ&pg=PA1">pp. 1–11</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161201074612/https://books.google.com/books?id=FzYVAAAAQAAJ&pg=PA1">Archived</a> 2016-12-01 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>: <i>"Propositio I. Lumen propagatur seu diffunditur non solum directe, refracte, ac reflexe, sed etiam alio quodam quarto modo, diffracte."</i> (Proposition 1. Light propagates or spreads not only in a straight line, by refraction, and by reflection, but also by a somewhat different fourth way: by diffraction.) On p. 187, Grimaldi also discusses the interference of light from two sources: <i>"Propositio XXII. Lumen aliquando per sui communicationem reddit obscuriorem superficiem corporis aliunde, ac prius illustratam."</i> (Proposition 22. Sometimes light, as a result of its transmission, renders dark a body's surface, [which had been] previously illuminated by another [source].)</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJean_Louis_Aubert1760" class="citation book cs1">Jean Louis Aubert (1760). <a rel="nofollow" class="external text" href="https://archive.org/details/memoirespourlhi146aubegoog"><i>Memoires pour l'histoire des sciences et des beaux arts</i></a>. Paris: Impr. de S. A. S.; Chez E. Ganeau. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/memoirespourlhi146aubegoog/page/n151">149</a>. <q>grimaldi diffraction 0–1800.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Memoires+pour+l%27histoire+des+sciences+et+des+beaux+arts&rft.place=Paris&rft.pages=149&rft.pub=Impr.+de+S.+A.+S.%3B+Chez+E.+Ganeau&rft.date=1760&rft.au=Jean+Louis+Aubert&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmemoirespourlhi146aubegoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiffraction" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSir_David_Brewster1831" class="citation book cs1">Sir David Brewster (1831). <a rel="nofollow" class="external text" href="https://archive.org/details/atreatiseonopti00brewgoog"><i>A Treatise on Optics</i></a>. London: Longman, Rees, Orme, Brown & Green and John Taylor. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/atreatiseonopti00brewgoog/page/n113">95</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Treatise+on+Optics&rft.place=London&rft.pages=95&rft.pub=Longman%2C+Rees%2C+Orme%2C+Brown+%26+Green+and+John+Taylor&rft.date=1831&rft.au=Sir+David+Brewster&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fatreatiseonopti00brewgoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiffraction" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Letter from James Gregory to John Collins, dated 13 May 1673. Reprinted in: <i>Correspondence of Scientific Men of the Seventeenth Century …</i>, ed. Stephen Jordan Rigaud (Oxford, England: <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>, 1841), vol. 2, pp. 251–255, especially <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0h45L_66bcYC&pg=PA254">p. 254</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161201061930/https://books.google.com/books?id=0h45L_66bcYC&pg=PA254">Archived</a> 2016-12-01 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThomas_Young1804" class="citation journal cs1">Thomas Young (1 January 1804). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7AZGAAAAMAAJ&pg=PA1">"The Bakerian Lecture: Experiments and calculations relative to physical optics"</a>. <i><a href="/wiki/Philosophical_Transactions_of_the_Royal_Society_of_London" class="mw-redirect" title="Philosophical Transactions of the Royal Society of London">Philosophical Transactions of the Royal Society of London</a></i>. <b>94</b>: 1–16. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1804RSPT...94....1Y">1804RSPT...94....1Y</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstl.1804.0001">10.1098/rstl.1804.0001</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:110408369">110408369</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society+of+London&rft.atitle=The+Bakerian+Lecture%3A+Experiments+and+calculations+relative+to+physical+optics&rft.volume=94&rft.pages=1-16&rft.date=1804-01-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A110408369%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1098%2Frstl.1804.0001&rft_id=info%3Abibcode%2F1804RSPT...94....1Y&rft.au=Thomas+Young&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D7AZGAAAAMAAJ%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiffraction" class="Z3988"></span>. (Note: This lecture was presented before the Royal Society on 24 November 1803.)</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Fresnel, Augustin-Jean (1816), "Mémoire sur la diffraction de la lumière" ("Memoir on the diffraction of light"), <i>Annales de Chimie et de Physique</i>, vol. 1, pp. 239–81 (March 1816); reprinted as "Deuxième Mémoire…" ("Second Memoir…") in <i>Oeuvres complètes d'Augustin Fresnel</i>, vol. 1 (Paris: Imprimerie Impériale, 1866), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1l0_AAAAcAAJ&pg=PA89">pp. 89–122</a>. (Revision of the <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1l0_AAAAcAAJ&pg=PA9">"First Memoir"</a> submitted on 15 October 1815.)</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Fresnel, Augustin-Jean (1818), "Mémoire sur la diffraction de la lumière" ("Memoir on the diffraction of light"), deposited 29 July 1818, "crowned" 15 March 1819, published in <i>Mémoires de l'Académie Royale des Sciences de l'Institut de France</i>, vol. <span class="serif-fonts" style="font-family: 'Georgia Pro', Georgia, 'DejaVu Serif', Times, 'Times New Roman', FreeSerif, 'DejaVu Math TeX', 'URW Bookman L', serif;">V</span> (for 1821 & 1822, printed 1826), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zNo-AQAAMAAJ&pg=PA339">pp. 339–475</a>; reprinted in <i>Oeuvres complètes d'Augustin Fresnel</i>, vol. 1 (Paris: Imprimerie Impériale, 1866), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1l0_AAAAcAAJ&pg=PA247">pp. 247–364</a>; partly translated as <a rel="nofollow" class="external text" href="https://archive.org/details/wavetheoryofligh00crewrich/page/80">"Fresnel's prize memoir on the diffraction of light"</a>, in H. Crew (ed.), <i>The Wave Theory of Light: Memoirs by Huygens, Young and Fresnel</i>, American Book Company, 1900, pp. 81–144. (First published, as extracts only, in <i>Annales de Chimie et de Physique</i>, vol. 11 (1819), pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SSRQAAAAcAAJ&pg=PA246">246–96</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SSRQAAAAcAAJ&pg=PA337">337–78</a>.)</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Christiaan Huygens, <a rel="nofollow" class="external text" href="https://archive.org/details/bub_gb_X9PKaZlChggC"><i>Traité de la lumiere</i> …</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160616191659/https://books.google.com/books?id=X9PKaZlChggC&pg=PP5">Archived</a> 2016-06-16 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> (Leiden, Netherlands: Pieter van der Aa, 1690), Chapter 1. From <a rel="nofollow" class="external text" href="https://archive.org/details/bub_gb_X9PKaZlChggC/page/n94">p. 15</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161201064555/https://books.google.com/books?id=X9PKaZlChggC&pg=PA15">Archived</a> 2016-12-01 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>: <i>"J'ay donc monstré de quelle façon l'on peut concevoir que la lumiere s'etend successivement par des ondes spheriques, … "</i> (I have thus shown in what manner one can imagine that light propagates successively by spherical waves, … ) (Note: Huygens published his <i><a href="/wiki/Treatise_on_Light" title="Treatise on Light">Traité</a></i> in 1690; however, in the preface to his book, Huygens states that in 1678 he first communicated his book to the French Royal Academy of Sciences.)</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">Baker, B.B. & Copson, E.T. (1939), <i>The Mathematical Theory of Huygens' Principle</i>, Oxford, pp. 36–40.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDietrich_Zawischa" class="citation web cs1">Dietrich Zawischa. <a rel="nofollow" class="external text" href="http://www.itp.uni-hannover.de/%7Ezawischa/ITP/spiderweb.html">"Optical effects on spider webs"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">21 September</span> 2007</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Optical+effects+on+spider+webs&rft.au=Dietrich+Zawischa&rft_id=http%3A%2F%2Fwww.itp.uni-hannover.de%2F%257Ezawischa%2FITP%2Fspiderweb.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiffraction" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArumugam2013" class="citation web cs1">Arumugam, Nadia (9 September 2013). <a rel="nofollow" class="external text" href="http://www.slate.com/blogs/browbeat/2013/09/09/iridescent_deli_meat_why_some_sliced_ham_and_beef_shine_with_rainbow_colors.html">"Food Explainer: Why Is Some Deli Meat Iridescent?"</a>. <i>Slate</i>. <a href="/wiki/The_Slate_Group" title="The Slate Group">The Slate Group</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130910021203/http://www.slate.com/blogs/browbeat/2013/09/09/iridescent_deli_meat_why_some_sliced_ham_and_beef_shine_with_rainbow_colors.html">Archived</a> from the original on 10 September 2013<span class="reference-accessdate">. Retrieved <span class="nowrap">9 September</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Slate&rft.atitle=Food+Explainer%3A+Why+Is+Some+Deli+Meat+Iridescent%3F&rft.date=2013-09-09&rft.aulast=Arumugam&rft.aufirst=Nadia&rft_id=http%3A%2F%2Fwww.slate.com%2Fblogs%2Fbrowbeat%2F2013%2F09%2F09%2Firidescent_deli_meat_why_some_sliced_ham_and_beef_shine_with_rainbow_colors.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiffraction" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAndrew_Norton2000" class="citation book cs1">Andrew Norton (2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=XRRMxjr24pwC&pg=PA102"><i>Dynamic fields and waves of physics</i></a>. CRC Press. p. 102. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7503-0719-2" title="Special:BookSources/978-0-7503-0719-2"><bdi>978-0-7503-0719-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dynamic+fields+and+waves+of+physics&rft.pages=102&rft.pub=CRC+Press&rft.date=2000&rft.isbn=978-0-7503-0719-2&rft.au=Andrew+Norton&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DXRRMxjr24pwC%26pg%3DPA102&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiffraction" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChiaoGarmireTownes1964" class="citation journal cs1">Chiao, R. Y.; Garmire, E.; Townes, C. H. (1964). <a rel="nofollow" class="external text" href="http://journals.aps.org/prl/pdf/10.1103/PhysRevLett.13.479">"Self-Trapping of Optical Beams"</a>. <i>Physical Review Letters</i>. <b>13</b> (15): 479–482. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1964PhRvL..13..479C">1964PhRvL..13..479C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.13.479">10.1103/PhysRevLett.13.479</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Self-Trapping+of+Optical+Beams&rft.volume=13&rft.issue=15&rft.pages=479-482&rft.date=1964&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.13.479&rft_id=info%3Abibcode%2F1964PhRvL..13..479C&rft.aulast=Chiao&rft.aufirst=R.+Y.&rft.au=Garmire%2C+E.&rft.au=Townes%2C+C.+H.&rft_id=http%3A%2F%2Fjournals.aps.org%2Fprl%2Fpdf%2F10.1103%2FPhysRevLett.13.479&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiffraction" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRahmat-Samii2013" class="citation journal cs1"><a href="/wiki/Yahya_Rahmat-Samii" title="Yahya Rahmat-Samii">Rahmat-Samii, Yahya</a> (June 2013). "GTD, UTD, UAT, and STD: A Historical Revisit and Personal Observations". <i>IEEE Antennas and Propagation Magazine</i>. <b>55</b> (3): 29–40. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013IAPM...55...29R">2013IAPM...55...29R</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FMAP.2013.6586622">10.1109/MAP.2013.6586622</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Antennas+and+Propagation+Magazine&rft.atitle=GTD%2C+UTD%2C+UAT%2C+and+STD%3A+A+Historical+Revisit+and+Personal+Observations&rft.volume=55&rft.issue=3&rft.pages=29-40&rft.date=2013-06&rft_id=info%3Adoi%2F10.1109%2FMAP.2013.6586622&rft_id=info%3Abibcode%2F2013IAPM...55...29R&rft.aulast=Rahmat-Samii&rft.aufirst=Yahya&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiffraction" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKouyoumjianPathak1974" class="citation journal cs1">Kouyoumjian, R. G.; Pathak, P. H. (November 1974). "A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface". <i><a href="/wiki/Proceedings_of_the_IEEE" title="Proceedings of the IEEE">Proceedings of the IEEE</a></i>. <b>62</b> (11): 1448–1461. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FPROC.1974.9651">10.1109/PROC.1974.9651</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+IEEE&rft.atitle=A+uniform+geometrical+theory+of+diffraction+for+an+edge+in+a+perfectly+conducting+surface&rft.volume=62&rft.issue=11&rft.pages=1448-1461&rft.date=1974-11&rft_id=info%3Adoi%2F10.1109%2FPROC.1974.9651&rft.aulast=Kouyoumjian&rft.aufirst=R.+G.&rft.au=Pathak%2C+P.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiffraction" class="Z3988"></span></span> </li> <li id="cite_note-JMC-23"><span class="mw-cite-backlink">^ <a href="#cite_ref-JMC_23-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-JMC_23-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">John M. Cowley (1975) <i>Diffraction physics</i> (North-Holland, Amsterdam) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-444-10791-6" title="Special:BookSources/0-444-10791-6">0-444-10791-6</a></span> </li> <li id="cite_note-Halliday-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-Halliday_24-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHallidayResnickWalker2005" class="citation cs2">Halliday, David; Resnick, Robert; Walker, Jerl (2005), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/isbn_0471216437"><i>Fundamental of Physics</i></a></span> (7th ed.), USA: John Wiley and Sons, Inc., <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-23231-5" title="Special:BookSources/978-0-471-23231-5"><bdi>978-0-471-23231-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamental+of+Physics&rft.place=USA&rft.edition=7th&rft.pub=John+Wiley+and+Sons%2C+Inc.&rft.date=2005&rft.isbn=978-0-471-23231-5&rft.aulast=Halliday&rft.aufirst=David&rft.au=Resnick%2C+Robert&rft.au=Walker%2C+Jerl&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fisbn_0471216437&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiffraction" class="Z3988"></span></span> </li> <li id="cite_note-Fowles1975-25"><span class="mw-cite-backlink">^ <a href="#cite_ref-Fowles1975_25-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Fowles1975_25-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrant_R._Fowles1975" class="citation book cs1">Grant R. Fowles (1975). <i>Introduction to Modern Optics</i>. Courier Corporation. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-65957-2" title="Special:BookSources/978-0-486-65957-2"><bdi>978-0-486-65957-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Modern+Optics&rft.pub=Courier+Corporation&rft.date=1975&rft.isbn=978-0-486-65957-2&rft.au=Grant+R.+Fowles&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiffraction" class="Z3988"></span></span> </li> <li id="cite_note-Hecht2002-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hecht2002_26-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHecht2002" class="citation book cs1">Hecht, Eugene (2002). <i>Optics</i> (4th ed.). United States of America: Addison Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8053-8566-3" title="Special:BookSources/978-0-8053-8566-3"><bdi>978-0-8053-8566-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Optics&rft.place=United+States+of+America&rft.edition=4th&rft.pub=Addison+Wesley&rft.date=2002&rft.isbn=978-0-8053-8566-3&rft.aulast=Hecht&rft.aufirst=Eugene&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiffraction" class="Z3988"></span></span> </li> <li id="cite_note-IchimiyaCohen2004-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-IchimiyaCohen2004_27-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAyahiko_IchimiyaPhilip_I._Cohen2004" class="citation book cs1">Ayahiko Ichimiya; Philip I. Cohen (13 December 2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=AUVbPerNxTcC"><i>Reflection High-Energy Electron Diffraction</i></a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-45373-8" title="Special:BookSources/978-0-521-45373-8"><bdi>978-0-521-45373-8</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170716041343/https://books.google.com/books?id=AUVbPerNxTcC">Archived</a> from the original on 16 July 2017.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Reflection+High-Energy+Electron+Diffraction&rft.pub=Cambridge+University+Press&rft.date=2004-12-13&rft.isbn=978-0-521-45373-8&rft.au=Ayahiko+Ichimiya&rft.au=Philip+I.+Cohen&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DAUVbPerNxTcC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiffraction" class="Z3988"></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeutzeWoutsvan_der_SpoelWeckert2000" class="citation journal cs1">Neutze, Richard; Wouts, Remco; van der Spoel, David; Weckert, Edgar; Hajdu, Janos (August 2000). <a rel="nofollow" class="external text" href="https://www.nature.com/articles/35021099">"Potential for biomolecular imaging with femtosecond X-ray pulses"</a>. <i>Nature</i>. <b>406</b> (6797): 752–757. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2000Natur.406..752N">2000Natur.406..752N</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F35021099">10.1038/35021099</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1476-4687">1476-4687</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/10963603">10963603</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:4300920">4300920</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Potential+for+biomolecular+imaging+with+femtosecond+X-ray+pulses&rft.volume=406&rft.issue=6797&rft.pages=752-757&rft.date=2000-08&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A4300920%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2000Natur.406..752N&rft.issn=1476-4687&rft_id=info%3Adoi%2F10.1038%2F35021099&rft_id=info%3Apmid%2F10963603&rft.aulast=Neutze&rft.aufirst=Richard&rft.au=Wouts%2C+Remco&rft.au=van+der+Spoel%2C+David&rft.au=Weckert%2C+Edgar&rft.au=Hajdu%2C+Janos&rft_id=https%3A%2F%2Fwww.nature.com%2Farticles%2F35021099&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiffraction" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChapmanCalemanTimneanu2014" class="citation journal cs1">Chapman, Henry N.; Caleman, Carl; Timneanu, Nicusor (17 July 2014). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4052855">"Diffraction before destruction"</a>. <i>Philosophical Transactions of the Royal Society B: Biological Sciences</i>. <b>369</b> (1647): 20130313. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstb.2013.0313">10.1098/rstb.2013.0313</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4052855">4052855</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/24914146">24914146</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society+B%3A+Biological+Sciences&rft.atitle=Diffraction+before+destruction&rft.volume=369&rft.issue=1647&rft.pages=20130313&rft.date=2014-07-17&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4052855%23id-name%3DPMC&rft_id=info%3Apmid%2F24914146&rft_id=info%3Adoi%2F10.1098%2Frstb.2013.0313&rft.aulast=Chapman&rft.aufirst=Henry+N.&rft.au=Caleman%2C+Carl&rft.au=Timneanu%2C+Nicusor&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4052855&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiffraction" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffraction&action=edit&section=21" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Diffraction" class="extiw" title="commons:Diffraction"><span style="font-style:italic; font-weight:bold;">Diffraction</span></a>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/I_30.html">The Feynman Lectures on Physics Vol. I Ch. 30: Diffraction</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.xtal.iqf.csic.es/Cristalografia/parte_05-en.html">"Scattering and diffraction"</a>. <i>Crystallography</i>. <a href="/wiki/International_Union_of_Crystallography" title="International Union of Crystallography">International Union of Crystallography</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Crystallography&rft.atitle=Scattering+and+diffraction&rft_id=https%3A%2F%2Fwww.xtal.iqf.csic.es%2FCristalografia%2Fparte_05-en.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiffraction" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=ZUszmEDm3FU&t=35s">Using a cd as a diffraction grating</a> at YouTube</li></ul> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box 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